La Regla del Producto de las Probabilidades

Igual que tenemos diferentes propiedades multiplicativas y de suma para los números reales, lo mismo ocurre con las probabilidades de los sucesos.

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    Cuando se trata de un solo suceso, no necesitamos buscar propiedades específicas para modelizar la propiedad del suceso. Pero si tenemos dos o más sucesos, entonces se vuelve realmente complicado modelizar las propiedades de dichos sucesos, y sobre todo relacionar estos sucesos. De ahí que existan Reglas de Probabilidad para relacionar distintos sucesos examinando sus propiedades.

    En tales casos, necesitamos relacionar dichos sucesos, así como sus intersecciones y uniones, exploremos más sobre estas propiedades, empezando por La regla de adición de la Probabilidad.

    La regla de adición

    Considera dos sucesos A y B, tales que formen parte del espacio muestral S. Sea PA{"x":[[255,255,255,255,255,255,255,255,255,255,255,255,255,255,255,255,256,258,260,263,266,270],[249,243,243,293,305,315,322,327,330,331,332,332,331,327,320,311,301,290,278,266,255,249,244,242],[443,435,428,419,408,397,387,378,369,361,356,354,353,353,353,356,361,367,375,384,394,404,415],[501,496,493,488,482,477,474,471,469,467,465,464,463,463,463,463,463,463,465,467,470,473,477,481,486,489,493,496,498,498,499,500,500,500,502,504,505,507,508,509,510,510,510,511,511,512,512,513,514,515,516,517,520,521],[464,468,475,484,493,502,511,517,522,527,529],[552,560,566,574,580,585,589,590,590,590,590,587,583,579,573,568,564,560,556,553,551]],"y":[[168,172,179,188,201,216,233,253,271,289,310,332,355,377,397,415,430,440,448,453,456,456],[205,197,192,156,152,151,151,152,155,159,164,169,175,183,193,203,214,226,236,245,252,257,259,260],[171,174,177,185,195,207,220,233,247,262,275,288,300,312,322,332,339,345,349,351,351,351,348],[214,218,225,235,247,258,269,279,288,296,303,307,310,311,310,307,303,294,282,272,261,248,236,225,216,210,205,202,201,200,200,200,202,205,209,216,224,234,246,259,272,285,296,305,315,322,329,334,338,341,342,343,344,344],[273,277,277,275,274,273,272,272,271,271,271],[186,189,193,201,211,222,234,248,262,277,291,306,321,336,351,366,379,391,401,409,415]],"t":[[0,8,17,21,32,39,48,56,65,73,82,89,98,106,115,123,132,140,149,156,165,171],[419,448,465,505,515,522,532,538,548,556,565,573,581,590,598,606,615,623,631,639,648,656,665,672],[1035,1055,1065,1075,1082,1090,1098,1107,1117,1122,1134,1139,1152,1156,1165,1173,1186,1189,1200,1211,1215,1223,1231],[1453,1464,1472,1482,1490,1503,1508,1515,1523,1533,1540,1553,1555,1567,1606,1614,1622,1634,1639,1651,1656,1667,1674,1683,1690,1703,1706,1715,1723,1734,1740,1757,1765,1772,1785,1789,1801,1806,1817,1822,1833,1840,1849,1856,1865,1873,1884,1889,1901,1907,1915,1922,1939,1947],[2151,2164,2174,2182,2190,2204,2207,2218,2223,2234,2239],[2461,2489,2499,2514,2523,2532,2533,2540,2549,2557,2569,2573,2585,2590,2599,2606,2615,2624,2635,2639,2649]],"version":"2.0.0"} yPB son las probabilidades de los sucesos A y B, respectivamente.

    La ley de la suma de probabilidades , también llamada regla de la suma o regla de la suma, establece que la probabilidad de que ocurran los dos sucesos que son la unión de A y B viene dada por

    PAB=PA+PB-P(AB)

    donde PAB denota la probabilidad de que ocurra A o B, y P(AB) denota la probabilidad de que ocurran A y B.

    Suponemos que A y B son sucesos no vacíos y que su intersección no es el conjunto vacío.

    Para entender cómo se obtiene la fórmula anterior, visualicemos los conjuntos A y B como parte de un diagrama de Venn.

    Recordemos que los diagram as de Venn son diagramas en los que los conjuntos y el espacio muestral se representan como figuras geométricas para comprender mejor sus uniones, complementos e intersecciones.

    Considera el siguiente diagrama de Venn.

    Reglas de probabilidad, El diagrama de Venn de dos sucesos A y B, StudySmarter El diagrama de Venn de dos sucesos A y B, StudySmarter Originals

    En el diagrama anterior, el rectángulo verde representa el espacio muestral, y los dos círculos azules representan los sucesos A y B respectivamente.

    Si queremos hallar la probabilidad del suceso "A o B", será su unión, y podemos utilizar el diagrama de Venn para ver cómo hacerlo.

    Si sumamos las probabilidades de A y B, su intersección se contará dos veces, en lugar de una. Por tanto, tenemos que restar la intersección de ambas. Esto nos da:

    PAB=PA+PB-PAB

    Para cualquier suceso, la palabra "y" implica su "intersección" y la palabra "o " implica su "unión".

    La regla de la suma también puede ampliarse a tres sucesos, a saber, A, B y C,

    PABC=PA+PB+PC-PAB-P(AC)-P(BC)+P(ABC)

    donde se puede considerar la misma idea de los diagramas de Venn para derivar la fórmula.

    Dados dos sucesos X e Y cuyas probabilidades de que ocurran son respectivamente 0,3 y 0,4. La probabilidad de que ocurran X e Y es 0,1, halla la probabilidad de que ocurra X o Y.

    Solución

    Las probabilidades de X e Y son PX=0.3{"x":[[68,67,67,68,68,68,68,68,68,67,67,67,66,65,65,65,64,64,64,64,64,64,64,65],[61,61,61,62,63,64,69,75,81,87,92,96,98,100,101,102,103,104,103,102,99,97,89,87,79,76,70,70,69],[183,181,179,177,176,172,168,163,160,150,147,139,137,132,131,131,133,134,138,140,145,151,157,165,172,180,188],[235,235,233,233,235,236,240,241,247,249,255,260,265,267,274,277,279,281,287,289,295,298],[301,298,296,294,292,289,284,282,275,270,264,259,257,251,250,247,246,245,245,245],[341,342,343,346,348,350,352,358,361,368,370,373,376,376,376,376,376,375,374,372,370,365,363,357,355,351],[429,430,431,434,436,439,444,447,452,456,461,463,467,469,470],[428,429,431,433,441,447,455,462,466,475,477],[535,535,535,534,530,529,526,525,524,524,525,525,527,533,535,540,543,546,551,556,561,563,567,570,572,573,573,572,570,568,563,561,559,556,554,552,551],[617,619,620,621],[688,688,689,690,691,692,696,699,702,706,710,715,717,722,722,724,724,723,722,720,717,715,712,711,710,709,709,709,710,713,715,721,724,730,732,736,738,738,738,738,736,734,732,728,723,718,708,703,697,691,685,682,680]],"y":[[181,181,186,193,204,210,229,235,242,256,263,277,297,303,314,320,330,334,345,348,354,355,357,356],[185,181,179,177,176,174,171,168,166,166,168,172,175,181,184,187,194,201,208,212,220,224,235,238,248,250,255,256,256],[181,179,178,178,178,180,183,187,190,203,208,226,232,252,266,279,290,294,301,305,310,314,316,318,318,317,315],[191,189,186,185,185,186,192,194,204,208,221,230,238,242,250,253,256,258,264,265,269,270],[184,184,185,189,192,200,210,216,232,243,254,264,269,282,286,294,297,302,303,304],[173,171,170,171,172,174,176,182,185,198,202,211,226,231,241,247,252,262,268,279,283,296,300,310,312,317],[224,223,223,223,223,223,223,223,223,223,224,224,225,226,226],[264,264,264,264,264,262,260,257,256,252,251],[207,208,210,213,223,227,240,245,257,261,273,276,282,287,288,288,288,287,285,281,276,273,267,260,252,249,239,232,226,224,219,218,217,216,215,215,215],[228,228,228,228],[201,194,190,188,187,186,182,180,178,176,175,175,175,177,178,184,186,192,196,201,205,207,211,213,215,218,219,221,223,224,225,227,228,232,234,239,243,247,249,254,259,261,264,269,273,277,281,281,281,281,279,277,275]],"t":[[0,8,24,31,41,48,58,64,65,73,81,91,98,108,114,115,124,132,141,148,160,164,174,189],[464,469,471,473,481,483,491,498,506,515,524,531,543,548,549,556,565,574,581,582,591,598,608,615,624,631,641,648,648],[992,1006,1010,1015,1015,1024,1031,1041,1048,1058,1065,1074,1081,1094,1098,1108,1115,1127,1131,1132,1141,1148,1159,1165,1179,1185,1191],[1449,1467,1470,1473,1491,1498,1508,1515,1525,1532,1545,1549,1557,1565,1575,1582,1582,1590,1598,1608,1615,1625],[1859,1874,1877,1878,1882,1891,1898,1908,1915,1925,1932,1942,1948,1962,1966,1975,1982,1994,1998,1999],[2250,2266,2269,2270,2274,2282,2283,2292,2299,2308,2315,2325,2332,2344,2349,2349,2362,2365,2366,2379,2382,2394,2399,2411,2415,2428],[2776,2790,2793,2795,2799,2809,2815,2816,2826,2833,2842,2849,2859,2865,2870],[3068,3083,3086,3091,3099,3109,3116,3125,3133,3142,3149],[3408,3424,3427,3428,3441,3449,3459,3466,3475,3482,3492,3499,3509,3516,3525,3532,3533,3541,3552,3561,3566,3566,3576,3583,3592,3599,3609,3616,3626,3632,3642,3649,3649,3659,3666,3666,3676],[3850,3858,3866,3874],[4254,4271,4274,4283,4283,4291,4299,4309,4316,4326,4333,4343,4349,4362,4366,4378,4383,4392,4399,4408,4416,4427,4433,4433,4441,4450,4450,4459,4467,4476,4483,4493,4499,4510,4517,4526,4533,4543,4552,4562,4566,4567,4576,4583,4593,4599,4609,4617,4626,4634,4643,4650,4654]],"version":"2.0.0"} y PY=0.4{"x":[[116,115,114,114,114,115,116,117,117,119,119,120,121,121,121,122,122,122,122,122,122,122,122,122,123],[100,98,95,94,94,99,101,109,113,119,122,128,134,139,141,146,147,148,148,145,141,135,133,128,124,123,121,119,118],[232,229,227,223,220,218,212,202,199,190,188,182,181,180,178,179,181,184,186,191,198,205,209,213,222,227],[268,270,273,275,280,283,286,292,294,299,301,302,306,308,310,315,317,321,325,330,334,336,340,344,347,350,353,355,358,359,360,360,359,358,357,355,352,349,345,343,339,337,335,330,329,325,322,321,320,319,319,319,319,321,322,323],[398,399,403,405,408,411,418,420,425,425,425,424,420,418,414,409,407,404,398,396,392,391,390],[494,493,492,491,491,493,494,500,503,511,514,517,524,527,529,532,537,541,544,545],[488,490,491,494,501,504,517,521,532,534,537,541],[648,646,643,641,639,637,636,634,633,633,634,638,640,647,652,659,662,666,669,677,684,693,695,700,705,706,706,706,704,703,701,694,692,683,681,678,672,669,665,663],[765],[841,841,841,840,839,837,836,833,832,830,827,823,819,816,815,817,818,821,825,831,833,836,841,846,852,858,860,866,872,876,881],[855,855,856,856,856,856,856,855,853,850,848,845,843,841,839,839,839,840]],"y":[[159,158,157,156,157,161,168,177,183,206,214,239,247,257,272,286,300,305,310,315,324,329,332,339,342],[187,183,179,174,170,164,162,156,155,154,154,155,158,162,164,172,175,185,193,202,211,219,223,229,234,235,236,238,238],[162,160,159,160,161,163,169,183,188,207,213,234,241,249,264,279,293,306,311,319,326,330,331,331,331,330],[170,171,174,176,184,187,191,202,206,213,216,218,221,222,223,223,223,222,220,215,210,208,201,194,188,181,173,170,161,159,154,153,153,153,153,155,160,166,175,179,188,193,199,213,220,233,246,255,259,271,273,281,283,288,289,289],[145,145,147,149,152,156,166,172,190,197,214,220,239,245,258,271,276,281,295,299,307,310,310],[191,191,190,190,189,189,189,189,189,189,189,189,188,188,188,188,188,188,189,189],[236,236,236,236,235,234,231,230,226,225,224,222],[174,180,186,196,202,213,218,234,239,253,257,267,270,275,276,276,275,274,273,268,263,251,247,238,224,214,204,200,192,189,186,179,177,175,175,175,176,177,178,180],[228],[146,148,150,154,161,168,172,179,182,186,192,200,208,216,221,223,224,225,226,227,227,227,227,226,225,224,223,222,221,220,218],[191,192,192,193,194,197,200,203,210,218,221,229,233,240,247,250,256,264]],"t":[[0,4,9,10,22,32,39,49,55,65,72,82,88,89,99,105,113,122,122,132,138,139,149,158,163],[394,402,416,422,432,439,449,455,466,472,472,482,489,499,506,515,522,532,538,548,555,565,572,580,589,589,598,605,606],[882,897,900,901,905,915,922,932,939,948,955,965,972,972,982,989,1000,1005,1016,1022,1034,1039,1039,1052,1055,1056],[1367,1367,1370,1373,1380,1381,1389,1399,1406,1415,1422,1423,1432,1439,1439,1449,1455,1465,1472,1482,1489,1489,1499,1506,1518,1522,1532,1539,1549,1557,1565,1572,1584,1589,1589,1602,1606,1619,1622,1634,1639,1639,1650,1655,1656,1668,1672,1682,1689,1699,1706,1715,1724,1734,1739,1748],[1961,1979,1982,1983,1989,1991,1999,2006,2016,2023,2032,2039,2049,2056,2065,2072,2073,2083,2089,2101,2106,2115,2123],[2395,2402,2407,2416,2439,2449,2456,2466,2472,2482,2489,2489,2499,2506,2506,2515,2523,2532,2539,2540],[2726,2748,2756,2759,2766,2773,2783,2789,2799,2806,2806,2816],[3133,3149,3152,3153,3157,3166,3174,3183,3189,3199,3206,3216,3224,3233,3239,3249,3256,3257,3267,3273,3283,3290,3300,3306,3320,3323,3334,3340,3349,3352,3359,3369,3373,3383,3389,3390,3400,3406,3407,3416],[3659],[3970,3986,3990,3991,3999,4006,4017,4023,4023,4033,4040,4050,4057,4069,4073,4086,4090,4100,4107,4118,4123,4124,4132,4140,4150,4157,4167,4173,4183,4190,4200],[4404,4409,4416,4423,4424,4434,4440,4440,4448,4457,4467,4473,4474,4484,4491,4500,4507,4517]],"version":"2.0.0"}.

    La probabilidad de que ocurran ambas es PXY=0.1{"x":[[142,142,142,142,142,142,143,143,144,144,145,146,146,148,148,149,150,151,151,152,153,153,154,155,155,155,155,155],[138,136,135,135,136,138,140,149,152,164,168,178,181,189,191,193,196,197,199,198,197,193,191,189,184,179,176,169,167,163],[283,273,269,266,264,258,255,249,247,245,243,242,242,242,243,244,246,247,252,256,262,269,272,282,286,298,301,305],[325,325,325,326,329,339,347,357,360,368,371,375,381,384,389,395,399,403,407,408,412,413],[399,398,397,396,394,393,392,388,385,382,375,363,359,349,346,340,339,337,335,334,335],[446,445,444,443,443,443,443,445,446,448,449,452,457,460,463,469,475,480,482,486,490,492,494,498,499,503,504,505,507,507,508,508,508,508,509,509,510,510,511,512],[542,543,544,549,551,553,556,558,559,561,562,563,566,568,570,574,576,578,580,582,583,587,588,590,591,591,592,592,592,591,591,591,591,591,591,591,590,590,589,588,587,585,585,584,584,583,583,583,583,583,583,583,584,584,585,586],[631,633,634,635,637,639,644,646,653,655,660,661,663,663,663,662,660,658,656,651,646,644,640,636,635,635],[734,736,738,740,744,747,750,756,765,771,776,779,786,787,792,793,793],[729,729,731,732,734,736,741,747,759,767,775,783,786,795,798,805,807,808,810],[878,877,877,877,875,875,873,871,870,870,870,870,871,872,874,877,880,884,887,892,897,900,903,909,912,915,924,930,935,940,942,944,944,944,943,940,939,937,928,921,914,911,900,897,894,892,888,885,884],[997,996,995,995],[1048,1048,1048,1048,1048,1048,1048,1048,1048,1047,1047,1047,1047,1047,1048,1049,1050,1052,1053,1053]],"y":[[166,167,170,174,177,186,198,210,224,232,254,262,270,287,295,302,316,323,329,334,344,349,353,363,368,371,372,371],[187,185,183,180,178,176,173,165,163,156,156,157,158,165,168,172,179,182,193,201,205,218,223,227,234,242,245,253,255,258],[161,167,173,177,183,196,203,225,232,241,256,264,272,279,291,296,301,306,315,320,326,329,330,330,329,324,321,318],[194,195,197,199,203,214,223,232,235,241,244,247,252,255,261,267,273,278,282,284,286,286],[185,185,184,184,184,185,186,190,194,198,210,230,236,255,260,273,276,281,287,290,291],[272,271,270,267,266,258,255,244,235,225,220,210,200,195,190,183,178,176,176,177,181,186,189,198,201,210,213,216,223,227,231,240,244,252,260,265,270,272,272,272],[170,171,173,182,186,190,199,207,210,214,216,218,220,221,221,220,218,214,209,203,199,188,184,174,171,169,166,165,164,164,165,167,168,173,180,188,193,204,215,228,234,252,257,273,276,286,288,290,293,294,295,294,293,289,287,285],[135,134,134,134,135,137,141,145,161,166,185,192,209,222,234,247,260,266,272,285,298,303,312,321,323,322],[196,196,196,196,195,195,195,195,195,195,195,195,195,195,196,196,197],[231,232,232,232,232,232,231,230,228,226,225,225,224,224,224,224,224,224,223],[184,184,186,187,192,196,206,217,228,237,246,250,261,264,267,272,276,278,279,280,280,279,278,275,273,270,261,253,245,235,226,218,214,205,202,197,195,194,190,188,188,188,189,190,190,190,191,191,192],[233,235,236,237],[180,189,198,204,210,216,229,235,242,253,265,270,281,289,296,304,308,311,312,313]],"t":[[0,6,11,12,13,22,31,38,48,54,64,71,71,81,87,88,98,104,104,114,121,121,131,138,148,154,164,179],[583,599,602,609,613,613,623,631,638,647,655,663,671,681,688,688,696,704,714,721,731,738,747,749,754,764,772,781,788,797],[1222,1242,1247,1255,1258,1263,1271,1281,1288,1288,1298,1305,1305,1314,1321,1322,1330,1334,1338,1347,1356,1366,1372,1382,1388,1399,1405,1410],[3792,3795,3799,3806,3814,3822,3832,3839,3849,3856,3856,3865,3872,3873,3882,3889,3899,3907,3915,3923,3935,3951],[4135,4141,4144,4148,4156,4156,4166,4172,4173,4181,4189,4199,4206,4217,4222,4232,4239,4250,4256,4266,4272],[4662,4669,4673,4682,4690,4699,4706,4716,4723,4732,4739,4749,4756,4756,4766,4773,4782,4789,4790,4799,4806,4816,4823,4832,4840,4849,4856,4856,4866,4873,4873,4883,4889,4899,4906,4916,4923,4933,4940,4956],[5267,5284,5291,5299,5306,5306,5316,5323,5333,5340,5343,5350,5356,5367,5373,5384,5390,5402,5406,5416,5423,5432,5441,5451,5456,5456,5466,5473,5473,5498,5506,5507,5516,5523,5535,5539,5540,5551,5556,5569,5573,5586,5590,5599,5610,5619,5623,5623,5633,5640,5649,5656,5666,5673,5674,5681],[5953,5963,5967,5968,5973,5975,5983,5990,6000,6007,6018,6024,6032,6040,6048,6056,6066,6073,6073,6086,6090,6100,6106,6116,6124,6133],[6662,6669,6675,6684,6690,6690,6700,6707,6717,6724,6733,6741,6750,6757,6767,6782,6782],[7234,7243,7265,7274,7274,7284,7290,7305,7308,7317,7324,7333,7340,7350,7357,7367,7373,7374,7384],[8443,8466,8474,8484,8491,8491,8501,8509,8517,8524,8534,8541,8551,8559,8562,8569,8574,8584,8591,8600,8607,8608,8618,8624,8625,8633,8641,8651,8658,8667,8674,8684,8692,8701,8708,8716,8716,8724,8734,8741,8751,8757,8768,8774,8775,8785,8791,8801,8807],[9061,9065,9074,9091],[9467,9483,9486,9487,9491,9494,9502,9508,9508,9518,9525,9535,9541,9552,9558,9568,9575,9584,9592,9601]],"version":"2.0.0"}.

    Se nos pide hallar la probabilidad de X o Y, que no es sino la probabilidad de su uniónPXY.

    Por tanto, podemos utilizar la regla de la suma para hallarla,

    PXY=PX+PY-PXY

    Sustituyendo los valores adecuados, obtenemos

    PXY=0.3+0.4-0.1=0.6

    Por tanto, la probabilidad de que ocurra el suceso X o Y es 0,6.

    Regla de adición para sucesos disjuntos

    A veces puede darse el caso de que dos sucesos no tengan nada que ver entre sí cuando su intersección es un conjunto nulo.

    Dos sucesos, A y B, se llaman S ucesos disjuntos si su intersección es un conjunto nulo, es decir

    AB=O

    Ahora, para hallar la probabilidad de la unión de dos sucesos disjuntos, utilizamos la regla de adición

    PAB=PA+PB-PAB{"x":[[24,24,24,24,24,24,25,25,25,25,25,25,24,24,24,24,24,24,24,24,25,25,26,27],[23,22,21,20,19,18,18,22,26,32,37,39,47,49,55,56,60,61,61,60,58,55,53,51,45,43,40,38],[92,91,90,88,86,83,82,78,76,75,75,76,77,78,79,82,86,89,93,95,97,99],[108,108,108,109,110,110,112,113,114,115,117,118,120,123,126,128,130,133,135,138,139,141,142,144,145,145,145,145,145,145,144,143,143,142,142,142,142],[116,114,113,114,115,116,118,124,126,133,136,138],[183,182,182,182,181,181,181,180,180,180,181,183,185,187,189,192,193,195,199,202,205,207,211,215,216,219,220,221,221,221,222,222,222,222],[249,248,248,248,248,248,248,248,247,246,245,244,243,243,243,243],[248,248,247,247,248,250,253,257,258,264,266,271,272,274,275,275,275,275,275,274,273,272,269,265,262,260,259,258,258,260,261,265,267,271,272,274,276,278,278,278,278,278,277,276,272,269,267,260,258,251,249,245],[294,295,299,300,302,307,310,314,315,316,316,317,316,315,313,309,305,301,298,296,293,292,291,291],[349,350,351,353,356,361,367,370,378,380,386,387,389,389,390],[341,340,341,343,344,350,353,360,363,369,371,373,376,379],[426,425,425,424,423,423,422,422,422,422,422,422,422,422,422,422,423,423,424],[423,422,422,423,424,425,427,432,437,443,446,449,451,451,451,450,450,447,445,443,439,437,433,428,425,424,423,423],[484,482,479,476,474,467,462,458,455,454,454,454,457,458,464,466,473],[507,507,506,506,506,506,506,507,509,510,514,517,520,522,524,528,530,532,534,537,539,541,542,543,543,543,545,545,547,548,550,550,550,550,550,549,549,549],[511,509,508,508,509,511,514,517,520,522,524,526,528],[563,564,564,566,566,570,571,574,576,577,577,575,574,573,572],[623,622,621,622,625,629,633,636,645,647,655,657,661,662,663],[643,642,642,641,642,642,642,642,642,642,642,640,640,640,640],[684,684,684,684,684,684,681,680,678,676,675,674,673,673,673,674,674,675,676,677,679,680],[686,686,686,686,686,687,690,694,698,703,708,710,715,716,715,714,709,707,700,696,691,688,686,682,680,679],[755,754,753,752,749,746,744,741,736,734,731,729,727,727,727,730,731,737,740,748,753,756],[785,786,786,787,787,787,787,786,785,785,785,784,784,784,784,784,784],[791,791,791,791,791,792,795,796,801,803,807,812,814,815,815,815,814,812,811,810,806,802,801,797,796,794,794,794,794,795,797,800,803,806,808,811,812,813,815,816,817,818,818,816,815,813,810,808,803,801,795,792,789,788],[833,832,831,831,832,833,834,836,838,839,841,843,844,844,844,843,841,840,836,835,830,829,826],[876,877,878,879,880,881,888,891,899,903,907,909,910],[925,926,927,927,927,927,927,926,926,925,925,924,924,924,923,923,923,923,923,924],[930,930,931,932,935,938,941,946,948,953,955,957,957,957,955,953,949,947,943,941,939,936,934],[990,989,988,986,982,978,973,968,965,964,962,961,961,961,962,963,967,968,973,976,979],[1001,1001,1000,1000,999,999,998,998,999,1000,1001,1003,1005,1007,1010,1014,1017,1020,1022,1025,1026,1028,1029,1030,1031,1032,1034,1034,1036,1037,1038,1039,1040,1040,1041,1041,1040,1039,1037,1036,1034,1034,1033],[1005,1003,1002,1003,1004,1005,1008,1011,1018,1021,1028,1033,1035,1039],[1058,1058,1058,1057,1057,1057,1057,1057,1057,1060,1061,1066,1070,1072,1079,1081,1085,1085,1088,1088,1089,1089,1089,1089,1090,1090,1090,1091,1092,1092,1092,1092,1091,1091,1090,1090,1090,1090,1091,1092],[1107,1108,1109,1110,1110,1111,1111,1111,1109,1108,1105,1105,1103,1103,1103,1104,1104,1105,1106,1107],[1107,1107,1107,1108,1110,1111,1114,1116,1119,1120,1122,1124,1127,1128,1128,1128,1128,1127,1126,1125,1122,1119,1118,1114,1113,1112,1111,1113,1114,1115,1116,1119,1120,1120,1121,1122,1122,1122,1122,1122,1121,1119,1117,1116,1112,1111,1108,1106],[1127,1128,1131,1133,1135,1137,1138,1140,1141,1141,1141,1141,1140,1138,1136,1134,1131,1129,1124,1122,1119,1114,1112,1110,1106,1103]],"y":[[302,303,304,311,322,328,347,353,369,374,392,403,415,421,425,430,442,448,450,451,451,450,446,440],[318,318,317,317,316,315,314,310,307,304,302,301,301,301,304,306,311,315,319,324,327,333,336,339,347,350,353,355],[319,319,319,321,326,335,341,361,375,387,398,403,417,421,426,430,432,432,431,429,428,426],[394,393,391,388,384,382,373,370,359,356,344,339,334,326,320,317,315,314,314,314,316,318,324,337,347,357,363,376,381,393,399,405,410,413,414,415,414],[362,362,362,361,361,361,360,358,357,355,354,353],[333,334,335,336,339,342,353,358,372,380,387,393,397,399,401,401,401,401,400,399,398,397,394,390,388,381,379,371,368,359,353,347,345,343],[339,338,337,336,338,340,346,350,364,374,380,391,394,401,402,404],[345,344,341,340,339,337,334,331,329,327,326,326,327,329,331,333,335,336,337,340,342,343,347,353,357,360,362,364,365,366,366,368,368,371,372,373,376,379,381,382,385,388,389,390,394,396,397,401,402,404,404,404],[298,298,297,298,300,308,315,324,329,333,338,353,362,372,378,392,401,409,416,420,425,427,429,430],[353,353,353,353,353,352,351,350,349,349,348,348,348,349,349],[379,379,379,379,379,379,378,377,376,374,374,373,372,372],[310,312,314,321,333,346,352,370,377,392,398,412,415,419,424,426,427,428,429],[328,327,324,322,321,320,318,316,314,314,314,316,319,323,325,328,331,336,339,342,347,350,354,357,360,361,362,363],[323,322,326,333,338,357,371,383,396,407,415,421,425,427,430,430,431],[412,410,408,407,405,399,392,388,375,370,353,343,334,331,327,322,320,318,317,316,317,320,329,337,346,350,362,366,377,380,389,391,398,403,406,409,410,409],[369,370,370,371,371,371,371,370,369,368,367,366,365],[308,308,307,308,310,319,324,341,353,365,374,383,393,397,400],[360,360,359,359,359,358,358,357,356,356,355,355,355,355,355],[343,341,340,340,341,344,348,353,359,366,369,380,382,387,388],[309,308,309,310,317,322,343,351,371,384,390,400,409,413,419,421,425,426,428,428,428,426],[327,326,324,323,321,320,318,317,316,315,315,315,320,324,332,334,342,345,352,355,358,360,361,363,364,364],[325,325,324,324,324,327,329,335,344,349,359,364,378,386,392,398,400,403,403,403,401,400],[325,326,330,333,347,352,366,374,382,384,387,393,395,398,399,402,403],[334,333,331,330,328,327,325,324,322,321,321,321,323,325,327,331,336,340,342,344,351,355,356,361,362,364,365,366,367,367,368,369,370,372,373,375,376,378,379,382,386,389,393,395,399,400,404,405,407,408,409,409,407,407],[309,308,308,307,309,309,311,315,320,323,327,336,341,358,367,375,383,387,396,399,406,408,411],[363,363,364,364,364,364,363,363,361,360,359,359,359],[326,326,327,328,336,340,350,355,365,375,383,391,397,400,407,409,413,414,415,414],[335,328,327,326,325,324,323,323,323,324,324,328,330,335,339,342,346,348,351,352,354,355,356],[320,321,322,324,328,334,342,349,357,361,371,375,378,383,387,389,395,396,398,399,399],[389,388,387,386,384,382,379,377,371,365,362,356,353,346,339,332,327,324,323,322,322,325,328,333,339,345,351,354,360,363,365,368,373,376,380,383,384,387,389,389,389,388,386],[358,358,358,358,358,358,358,358,356,356,353,352,351,349],[368,366,365,363,361,359,356,354,352,345,343,336,331,329,325,324,324,325,328,329,335,337,339,344,347,350,352,359,361,366,369,373,377,379,381,382,384,385,385,385],[326,325,325,325,326,326,331,333,343,347,358,361,369,373,376,378,380,381,381,381],[338,337,333,332,330,328,327,327,326,326,326,327,329,331,334,335,337,341,343,345,349,353,354,358,359,361,361,362,362,363,363,366,368,369,371,374,375,378,383,385,387,388,389,390,390,390,389,387],[295,294,293,294,296,300,306,314,323,332,337,350,355,368,377,385,393,397,402,405,408,412,414,415,417,418]],"t":[[0,16,18,25,36,42,51,58,68,75,84,91,102,108,110,117,125,136,142,150,153,158,167,175],[418,422,425,426,435,442,451,459,473,476,484,491,505,509,520,525,535,542,555,558,567,575,576,585,592,601,608,620],[995,1003,1018,1025,1038,1042,1051,1059,1068,1075,1085,1092,1101,1108,1118,1125,1135,1142,1153,1158,1159,1171],[1506,1512,1518,1525,1535,1543,1556,1560,1569,1575,1585,1592,1592,1604,1609,1618,1622,1626,1637,1642,1642,1651,1659,1669,1676,1685,1693,1703,1709,1722,1725,1736,1742,1755,1759,1760,1775],[2000,2012,2015,2037,2042,2043,2052,2059,2069,2075,2085,2088],[2851,2856,2859,2862,2869,2876,2886,2892,2902,2910,2919,2926,2938,2942,2952,2959,2969,2970,2977,2988,2993,3002,3011,3021,3027,3036,3043,3052,3059,3071,3076,3085,3093,3097],[3369,3377,3393,3402,3435,3436,3443,3452,3460,3471,3476,3486,3493,3505,3510,3519],[3723,3731,3734,3743,3743,3752,3759,3769,3776,3786,3793,3806,3810,3821,3826,3838,3843,3843,3852,3860,3860,3870,3876,3888,3893,3905,3910,3922,3927,3938,3944,3953,3960,3973,3976,3977,3989,3993,4003,4012,4021,4026,4027,4037,4043,4053,4060,4070,4076,4086,4093,4103],[4366,4381,4384,4385,4393,4402,4410,4420,4426,4427,4437,4443,4459,4460,4470,4476,4486,4494,4505,4510,4519,4520,4527,4531],[5003,5060,5068,5077,5086,5094,5103,5111,5122,5127,5139,5143,5154,5160,5170],[5374,5388,5394,5403,5410,5420,5427,5437,5444,5458,5460,5463,5471,5477],[5819,5826,5831,5835,5844,5854,5860,5871,5877,5887,5894,5907,5910,5911,5921,5927,5927,5936,5944],[6150,6161,6164,6165,6170,6178,6180,6188,6194,6204,6210,6221,6227,6238,6244,6244,6254,6260,6261,6269,6277,6278,6288,6294,6304,6311,6313,6319],[6657,6671,6686,6694,6704,6711,6720,6727,6738,6744,6754,6762,6771,6778,6789,6794,6808],[7159,7171,7178,7178,7188,7194,7205,7211,7221,7228,7237,7244,7253,7261,7261,7274,7278,7278,7290,7294,7304,7311,7323,7328,7339,7344,7357,7362,7373,7378,7389,7394,7407,7411,7423,7428,7441,7455],[7644,7649,7661,7678,7688,7694,7705,7711,7722,7728,7728,7737,7744],[8067,8078,8088,8105,8111,8122,8128,8138,8145,8154,8161,8171,8178,8187,8193],[8627,8634,8653,8678,8688,8695,8707,8712,8724,8729,8738,8745,8755,8761,8775],[8915,8928,8934,8937,8962,8974,8979,8990,8995,9006,9011,9027,9029,9039,9045],[9287,9300,9312,9315,9322,9328,9339,9345,9356,9362,9371,9378,9389,9389,9396,9409,9412,9412,9423,9429,9438,9446],[9645,9650,9653,9654,9662,9672,9678,9689,9695,9706,9712,9723,9738,9745,9758,9762,9775,9779,9791,9795,9805,9812,9813,9823,9829,9838],[10100,10105,10108,10112,10120,10129,10139,10145,10156,10158,10162,10173,10179,10189,10195,10206,10212,10223,10229,10241,10245,10255],[10478,10495,10504,10512,10522,10529,10541,10545,10558,10562,10563,10576,10579,10579,10592,10595,10596],[10788,10792,10796,10797,10805,10812,10823,10829,10840,10848,10856,10869,10873,10879,10889,10896,10905,10912,1091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    Observando que A y B son sucesos disjuntos, tenemos

    P(AB)=PO=0

    Introduciendo ahora el valor de su intersección, tenemos

    PAB=PA+PB{"x":[[54,54,53,53,54,54,54,54,53,52,52,52,52,52,52,52,52,52,53,53,54,55,56,56,58,59],[58,57,55,54,55,58,62,67,73,80,86,91,96,104,105,105,105,102,100,95,89,85,77,73,66,58,53,47,43,42,41],[185,177,174,172,167,160,156,145,141,132,130,126,125,125,127,129,133,135,141,150,157,165,172],[211,208,206,206,205,203,201,200,198,197,195,193,192,191,190,189,188,188,188,189,190,193,195,200,205,208,217,219,225,227,229,233,237,239,240,241,242,242,243,243,242,242,240,239,238,238,238,238,238,238,238,238],[200,198,197,200,204,209,215,218,227,232,236,237,239,241,242,243,244,244,243],[278,277,276,275,274,274,274,274,276,277,281,284,287,293,297,302,304,310,312,315,318,322,324,329,330,332,334,335,335,336,336,336,335,334,333,332],[369,370,370,371,371,371,370,369,368,367,366,365,365,364,364],[370,370,372,373,375,378,382,386,389,391,397,399,401,402,403,402,401,399,398,397,393,389,383,380,379,378,378,380,381,383,387,389,391,396,400,408,412,417,419,420,422,422,420,416,411,408,406,400,397,394,391,386,384,382],[429,432,434,443,448,454,458,461,463,464,464,464,462,457,452,447,445,439,437,433,433,432],[509,506,507,510,512,518,521,529,536,541,546,549,552,553],[510,511,512,516,521,527,534,537,546,548,554,555,558],[607,607,608,608,609,609,609,610,610,611,611,611,611,611,611,610,610,609,609,609,609,609,609,610],[618,616,617,618,622,625,628,635,638,645,647,652,654,655,656,656,654,650,645,639,634,628,623,619,617],[697,696,692,688,685,681,678,676,672,670,666,665,663,664,666,670,672,679,682,688,691,694,697],[718,720,722,724,725,727,729,732,733,738,740,743,744,746,748,749,750,751,751,751,752,753,753,753,754,755,755,755,756,756,757,758,759,760,761,762],[717,716,716,718,720,727,733,740,745,750],[787,788,792,793,794,796,800,803,804,807,807,807,807,806,804,802,800,797,793,788,786,783,781,780],[839,840,841,845,846,851,854,862,865,872,877,879,881,882],[864,865,866,867,867,867,866,864,862,860,859,856,855,854,854,854],[925,924,923,923,923,922,922,922,922,921,921,920,920,919,919,919,919,919,920,920,921],[928,929,932,935,938,943,948,956,960,964,966,966,966,965,962,960,955,951,947,945,943,939,937],[1003,1003,1002,1002,1001,999,997,994,990,987,985,983,981,981,981,983,988,992,994,1002,1005,1007,1009],[1028,1028,1028,1027,1026,1025,1024,1023,1023,1022,1022,1021,1021,1021,1021,1022,1022],[1025,1024,1024,1024,1025,1028,1029,1035,1039,1041,1045,1049,1052,1053,1053,1053,1050,1048,1045,1042,1041,1036,1034,1032,1031,1031,1034,1037,1040,1042,1047,1048,1052,1053,1054,1054,1053,1052,1050,1048,1047,1046,1042,1041,1037,1033,1029,1026,1024,1021,1020],[1077,1078,1085,1087,1088,1090,1092,1092,1092,1092,1089,1087,1080,1077,1067,1060,1053,1047,1045]],"y":[[340,338,337,338,340,350,355,376,391,406,422,437,454,461,482,488,497,501,507,512,515,516,516,515,509,506],[350,349,346,343,340,335,331,326,324,322,322,324,328,344,351,354,358,364,368,376,384,389,397,401,408,413,416,417,418,417,416],[339,340,341,342,347,355,359,374,380,399,405,423,430,445,455,463,470,472,476,478,478,476,471],[384,401,410,414,419,426,434,438,444,447,452,456,457,458,459,458,456,452,449,439,435,420,415,401,389,385,372,369,363,361,359,356,356,359,365,369,383,388,403,409,424,429,443,451,458,460,463,465,468,467,465,463],[408,408,408,408,408,407,406,405,402,400,399,399,399,399,399,399,400,401,402],[385,385,386,391,399,410,423,428,441,444,450,452,453,452,450,446,444,438,435,431,428,421,417,404,400,396,389,382,379,374,371,369,363,361,359,358],[369,369,370,373,380,390,400,405,415,423,434,440,442,449,452],[371,369,361,359,357,354,352,349,348,348,347,348,350,353,357,361,364,369,372,375,381,388,396,402,405,408,409,410,410,410,411,411,411,412,413,415,416,419,421,423,428,430,436,441,445,448,450,454,455,457,458,459,459,459],[339,340,341,348,354,363,372,381,390,395,410,415,426,441,451,461,465,477,479,485,486,486],[395,394,394,393,393,393,393,392,391,391,390,390,389,389],[427,428,428,428,427,425,424,423,420,420,419,419,419],[343,342,339,340,341,344,347,357,366,377,383,388,394,411,417,432,438,450,457,464,469,471,475,474],[354,349,348,347,346,345,345,346,346,350,351,356,358,360,364,368,373,378,384,389,394,398,402,404,404],[358,359,363,369,377,386,391,395,405,410,425,430,444,451,457,461,463,466,466,466,466,465,464],[432,426,421,416,410,403,395,387,383,373,370,365,363,360,359,358,360,362,369,372,382,386,389,397,405,412,416,423,429,432,436,439,441,443,444,444],[409,409,408,408,407,406,404,402,400,398],[348,347,345,345,345,346,349,353,356,368,372,386,391,396,404,413,417,425,432,442,444,449,450,452],[401,401,401,402,402,402,402,402,402,402,402,402,402,402],[381,381,380,380,381,384,389,396,403,411,415,425,430,434,436,438],[362,377,381,391,399,409,418,423,432,440,443,450,453,458,460,463,466,467,465,464,462],[366,365,362,361,361,360,360,361,363,366,370,372,379,382,388,390,395,397,400,401,401,403,403],[376,375,374,373,373,374,376,380,386,393,402,411,419,428,435,446,453,455,456,456,456,455,454],[375,376,378,383,391,400,409,417,420,428,431,437,440,442,443,443,442],[392,385,384,382,380,376,375,371,369,368,367,367,368,369,373,375,379,383,387,390,392,398,400,403,406,407,409,410,411,411,413,413,416,417,422,425,428,431,433,435,437,438,440,441,442,443,444,444,444,444,444],[357,358,367,370,374,382,391,395,409,414,427,432,445,449,461,466,470,474,474]],"t":[[0,8,13,22,29,39,46,56,63,75,80,91,98,105,113,122,130,131,141,148,158,163,171,179,189,196],[408,414,417,421,430,439,446,456,464,476,480,491,496,524,530,530,539,547,548,558,563,572,580,582,593,597,608,613,626,630,635],[999,1014,1022,1023,1030,1039,1047,1056,1063,1076,1080,1089,1097,1107,1113,1122,1130,1143,1148,1158,1164,1173,1180],[1535,1550,1556,1563,1564,1573,1580,1590,1597,1597,1607,1614,1623,1630,1631,1642,1648,1656,1663,1673,1680,1690,1697,1707,1714,1723,1730,1740,1747,1747,1756,1763,1775,1780,1791,1797,1809,1814,1825,1830,1840,1847,1859,1863,1873,1880,1881,1890,1897,1914,1925,1931],[2113,2119,2122,2155,2164,2173,2180,2190,2197,2207,2214,2223,2228,2230,2240,2247,2257,2264,2273],[2577,2591,2597,2607,2614,2624,2630,2640,2648,2657,2664,2675,2681,2693,2697,2707,2714,2723,2725,2730,2733,2740,2747,2757,2764,2764,2775,2780,2794,2797,2798,2807,2814,2824,2831,2840],[3037,3048,3051,3056,3064,3074,3081,3091,3097,3108,3114,3127,3131,3142,3150],[3327,3339,3343,3347,3348,3358,3364,3374,3381,3391,3397,3407,3415,3424,3432,3440,3448,3457,3460,3464,3476,3481,3493,3497,3510,3514,3526,3531,3540,3543,3548,3557,3558,3564,3573,3581,3591,3598,3606,3614,3622,3631,3644,3648,3657,3664,3665,3674,3681,3682,3695,3698,3699,3708],[3912,3927,3932,3941,3948,3956,3964,3974,3981,3991,3998,4010,4014,4026,4031,4041,4049,4062,4065,4076,4081,4091],[4396,4413,4439,4448,4458,4465,4475,4481,4491,4498,4508,4515,4529,4531],[4738,4755,4759,4765,4777,4781,4791,4798,4808,4815,4825,4831,4839],[5086,5100,5103,5115,5118,5123,5132,5141,5148,5161,5165,5165,5176,5181,5192,5198,5208,5215,5225,5232,5243,5249,5258,5287],[5498,5511,5516,5525,5532,5542,5548,5559,5565,5579,5582,5590,5598,5599,5608,5615,5625,5632,5645,5649,5661,5665,5677,5683,5690],[5962,5970,5973,5974,5982,5991,5998,5999,6009,6015,6027,6032,6041,6048,6059,6065,6074,6082,6092,6098,6099,6108,6113],[6466,6479,6490,6499,6509,6515,6528,6532,6542,6549,6561,6565,6566,6578,6583,6594,6610,6616,6625,6632,6641,6649,6649,6658,6665,6677,6682,6691,6699,6699,6711,6716,6728,6732,6743,6757],[6937,6952,6958,6966,6976,6982,6993,6999,7009,7016],[7302,7315,7318,7319,7324,7325,7333,7343,7349,7359,7366,7376,7382,7383,7393,7399,7412,7416,7426,7432,7444,7449,7449,7458],[7751,7769,7774,7782,7784,7794,7799,7810,7816,7826,7832,7842,7849,7859],[8029,8042,8045,8046,8051,8059,8066,8076,8083,8093,8099,8110,8116,8127,8133,8143],[8402,8416,8420,8425,8433,8443,8449,8458,8466,8476,8477,8483,8492,8500,8500,8513,8517,8528,8545,8550,8554],[8749,8762,8765,8766,8775,8783,8793,8800,8810,8816,8827,8833,8843,8850,8862,8867,8877,8883,8894,8900,8900,8910,8916],[9140,9150,9154,9162,9167,9177,9183,9194,9200,9210,9216,9227,9234,9244,9250,9275,9283,9293,9301,9313,9318,9319,9325],[9534,9546,9552,9560,9567,9577,9583,9594,9600,9613,9617,9627,9633,9643,9650,9660,9667],[9794,9809,9812,9817,9827,9833,9844,9850,9861,9867,9877,9883,9893,9900,9909,9917,9927,9933,9943,9950,9960,9967,9976,9983,9995,10000,10014,10017,10025,10033,10047,10050,10061,10067,10076,10083,10094,10100,10109,10117,10118,10128,10133,10134,10143,10150,10162,10167,10177,10184,10188],[10419,10434,10437,10442,10443,10450,10460,10467,10478,10484,10495,10500,10511,10517,10528,10534,10547,10550,10562]],"version":"2.0.0"}

    Se lanza dos veces y se anotan los resultados, halla la probabilidad de que el primer resultado sea 1 y el segundo resultado sea un número par.

    Se lanza un dado dos veces y se anotan los resultados, halla la probabilidad de que el primer resultado sea 1 y, el segundo resultado sea un número par.

    Solución

    Observa que el 1 no es un número par, por lo que los dos sucesos son disjuntos en este caso. La razón es que el resultado de que aparezca un número par no se solapa con el resultado de que aparezca 1 en el primer lanzamiento.

    Sean los dos sucesos A y B respectivamente,

    PA=16

    puesto que 1 es un resultado de 6 posibilidades, y,

    PB=36{"x":[[151,150,150,151,152,153,154,154,154,154,154,154,153,152,152,151],[143,144,146,149,163,168,179,183,191,197,202,204,207,207,206,205,199,196,193,181,178],[305,300,294,292,284,280,271,262,260,259,259,260,261,266,271,276,282,289,297,300,313],[342,343,343,343,343,344,344,345,346,346,347,347],[413,418,423,430,434,448,452,460,466,471,474,475,476,477,476,474,471,469,463],[537,549,552,564,568,576,579,586,589],[544,549,551,557,565,573,581,589,595],[684,684,684,685,688,701,704,707,710,717,720,723,724,725,724,722,717,715,712,708,707,707,706,706,706,707,710,711,717,719,724,725,728,728,728,727,721,719,710,707,704,701,691,683,679],[677,678,682,686,704,711,732,738,744,754,758,762,768,773,775,778,779,779],[747,743,741,735,732,725,721,717,715,713,713,713,716,717,721,723,725,729,731,735,736,740,741,742,742,738,736,728,726,723,717,712,709],[340,337,337,336,336,337,338,340,341,343,347,351,353,359,361,365,366,367,368,368,367,364,363,361,360,358,358,357,356,356,357,358,361,362,367,369,375,378,380,381,381,379,376,373,369,365,360,358,351,349,343,339,338]],"y":[[289,328,345,362,379,394,409,415,433,438,451,455,458,463,464,464],[286,275,273,269,260,258,254,253,253,255,259,261,269,273,284,289,303,308,312,329,332],[262,260,259,259,264,267,281,311,326,334,350,359,365,384,394,401,405,406,405,404,396],[284,284,283,285,287,301,307,327,333,342,347,352],[246,244,246,250,253,264,269,278,289,301,311,316,327,338,350,362,373,378,392],[285,282,281,279,278,276,275,273,271],[319,317,317,314,311,307,304,301,299],[244,236,232,230,226,216,214,213,212,209,208,209,210,214,218,223,233,236,241,248,250,252,256,257,258,260,262,262,265,266,270,271,276,279,283,285,290,292,296,297,297,298,297,296,295],[359,358,357,357,353,351,346,345,344,342,342,341,341,341,341,342,343,345],[393,395,396,403,406,418,427,437,446,456,460,466,473,473,474,474,473,471,469,465,462,453,450,442,440,437,437,438,440,442,446,451,454],[294,291,290,288,287,285,284,282,281,280,278,276,276,275,275,277,278,282,284,290,292,302,304,308,311,314,315,316,317,318,317,317,316,315,314,313,313,313,314,316,317,322,326,331,336,342,346,348,352,352,353,353,352]],"t":[[0,13,19,26,36,43,53,59,70,76,84,93,93,103,109,120],[429,442,446,451,460,470,476,477,487,493,503,510,520,526,536,543,554,560,560,579,585],[951,963,968,977,987,993,1004,1020,1027,1037,1043,1044,1052,1060,1068,1077,1087,1093,1103,1110,1120],[1461,1475,1478,1479,1481,1493,1502,1510,1520,1527,1527,1535],[2311,2324,2327,2335,2344,2354,2360,2371,2377,2387,2394,2394,2405,2411,2421,2427,2437,2444,2454],[2951,2964,2970,2977,2988,2994,2994,3004,3011],[3264,3277,3280,3286,3294,3304,3311,3321,3327],[3710,3722,3728,3738,3744,3762,3769,3770,3777,3788,3794,3805,3811,3821,3828,3838,3854,3861,3869,3878,3886,3889,3894,3903,3905,3911,3921,3928,3938,3944,3955,3962,3972,3978,3988,3994,4005,4011,4022,4028,4028,4039,4044,4055,4061],[4512,4532,4538,4545,4555,4561,4572,4578,4578,4589,4594,4595,4605,4611,4624,4628,4628,4644],[4977,4990,4996,5005,5011,5022,5028,5039,5045,5055,5061,5070,5078,5088,5095,5095,5106,5111,5112,5124,5128,5139,5145,5155,5161,5174,5178,5189,5195,5195,5207,5211,5220],[14609,14619,14625,14632,14642,14648,14659,14665,14665,14673,14682,14692,14698,14709,14715,14726,14732,14742,14748,14759,14765,14783,14790,14798,14809,14815,14816,14826,14832,14842,14859,14865,14876,14882,14892,14898,14909,14915,14926,14932,14942,14948,14959,14966,14976,14982,14992,14998,15009,15016,15026,15032,15040]],"version":"2.0.0"}

    ya que hay 3 números pares de entre las 6 posibilidades.

    Queremos hallar PAB. Así pues, utilizamos la fórmula de adición para sucesos disjuntos, ya que la ocurrencia de un suceso no afecta a la ocurrencia del otro. Por tanto, tenemos,

    PAB=PA+PB{"x":[[88,86,85,86,86,87,88],[88,88,89,89,91,96,112,120,124,127,129,134,135,135,133,130,126,121,114,109],[236,225,221,215,210,204,201,195,192,184,182,176,175,175,178,179,187,190,199,203,207],[263,263,263,263,264,266,271,273,274,278,282,286,288,292,296,297,300,300,301,301,302,302,302,302,302,302,303,304,305,306],[259,258,257,256,255,255,261,263,273,280,286,293,298,300,302,303,304],[348,348,346,346,349,350,354,358,360,366,368,376,382,386,391,395,399,400,404,405,406,406,406,406,405,404,403],[451,452,453,453,453,453,453,453,453,453,453],[450,449,449,451,452,454,462,466,474,477,479,484,485,488,488,486,485,481,479,475,469,465,462,460,460,461,463,468,472,477,480,488,490,494,496,499,501,502,500,497,495,490,487,485,475,470,466,465],[527,528,530,531,537,544,550,558,559,561,560,559,555,553,545,542,535,531,527],[620,620,622,624,633,636,649,656,662,664,667,669,670],[622,619,620,624,629,635,639,650,657,661,666,671,674,677],[716,716,717,717,718,719,719,719,718],[712,712],[713,710,709,710,712,719,721,730,736,742,744,746,748,750,750,749,748,743,738,732,727,724],[797,795,792,786,779,772,767,764,760,760,764,769,772],[809,808,809,811,814,816,824,830,832,838,839,842,843,843,842,840,840,839,838,837,836,836,836,836],[813,812,813,815,820,823,835,839,848,852,854,855],[881,883,884,886,888,891,893,898,899,901,900,898,897,891,889,886],[938,940,942,943,944,946,953,956,965,971,976,980,982,983,983],[960,962,963,963,963,962,961,961,960,960,961],[1019,1018,1018,1017,1016,1014,1011,1009,1004,1002,1000,999],[999,1000,1001],[1016,1015,1015,1016,1018,1023,1025,1031,1041,1046,1050,1052,1052,1049,1048,1041,1038,1030],[1075,1072,1070,1068,1066,1063,1061,1059,1056,1055,1054,1055,1056,1058,1063,1067,1069,1074],[1090,1090,1091,1091,1092,1092,1093,1093,1093,1092,1091,1090,1088,1087,1086,1087],[1095,1095,1095,1096,1098,1100,1105,1107,1112,1114,1118,1119,1119,1119,1119,1117,1115,1111,1108,1105,1103,1102,1102,1103,1106,1107,1111,1113,1114,1116,1118,1120,1121,1123,1123,1123,1123,1120,1117,1112,1107,1102,1099,1093,1092,1090],[1139,1138,1137,1136,1136,1136,1136,1138,1139,1141,1141,1138,1134,1130,1126,1123]],"y":[[228,393,416,420,421,420,418],[232,230,230,229,226,224,222,223,224,225,227,233,236,245,253,261,269,276,285,290],[251,254,255,258,263,271,275,285,291,312,319,340,347,364,372,376,383,385,391,392,393],[339,329,321,311,307,303,292,289,286,281,277,274,274,273,276,278,290,295,306,311,324,329,346,351,366,369,379,382,385,385],[317,317,317,317,317,316,313,312,309,307,305,304,303,303,303,304,304],[306,315,325,329,346,350,358,360,361,360,359,352,345,336,326,314,303,297,280,276,266,260,258,257,257,257,258],[263,263,264,265,275,280,300,307,323,328,333],[283,279,273,271,269,267,262,260,256,256,255,255,256,264,266,273,275,281,283,289,296,301,306,309,310,313,313,314,315,315,316,318,319,321,322,325,328,331,338,341,343,348,349,351,356,357,358,357],[241,241,241,242,246,254,264,286,291,309,321,327,345,351,367,373,384,389,394],[302,303,303,303,302,302,302,302,303,304,305,305,306],[331,332,332,332,332,331,330,328,327,326,326,326,326,326],[261,262,263,271,286,295,304,322,337],[381,386],[282,277,276,273,273,270,268,264,261,260,260,260,260,264,266,273,275,283,288,293,297,299],[275,276,278,282,289,299,311,317,333,338,350,355,357],[325,307,303,300,291,288,278,274,273,272,272,279,282,295,300,313,318,322,329,336,341,344,345,346],[311,312,311,311,309,307,303,301,299,298,298,298],[274,274,274,274,275,280,285,303,309,327,338,346,350,361,364,367],[311,310,309,308,307,307,303,302,299,298,298,298,298,298,299],[286,290,296,305,315,324,331,334,339,340,341],[278,279,280,285,295,307,321,327,344,354,362,364],[372,373,373],[279,277,276,273,272,270,269,268,267,267,268,270,274,279,282,290,292,299],[279,277,277,278,282,289,299,305,323,328,341,346,348,349,351,350,349,347],[293,292,293,294,295,300,306,314,318,327,331,333,336,337,336,335],[292,290,289,287,285,285,283,282,281,281,281,282,284,286,287,290,292,295,298,301,303,304,305,306,306,307,308,310,312,315,318,321,323,327,329,330,332,334,335,336,336,336,336,335,334,330],[278,282,287,296,301,305,310,322,326,336,342,349,354,359,364,365]],"t":[[0,16,30,33,40,58,65],[483,487,491,499,508,515,533,540,549,549,560,565,575,582,591,599,608,615,634,645],[1033,1034,1043,1049,1061,1066,1066,1075,1082,1092,1099,1108,1118,1124,1136,1144,1149,1157,1165,1175,1178],[1521,1528,1534,1538,1544,1549,1557,1566,1566,1574,1583,1592,1599,1609,1616,1624,1636,1645,1649,1650,1661,1666,1676,1683,1692,1699,1709,1716,1735,1742],[1955,1960,1970,1974,1982,1992,2008,2016,2024,2033,2041,2058,2066,2076,2082,2096,2099],[2422,2434,2438,2444,2463,2466,2476,2483,2492,2499,2509,2516,2527,2537,2544,2549,2561,2566,2576,2583,2593,2599,2609,2616,2617,2629,2641],[2841,2843,2858,2866,2875,2883,2892,2899,2909,2916,2920],[3152,3164,3169,3174,3175,3183,3195,3200,3208,3216,3217,3226,3234,3251,3262,3266,3275,3283,3284,3293,3300,3309,3316,3325,3334,3341,3353,3361,3366,3378,3383,3393,3400,3408,3410,3416,3426,3435,3451,3462,3466,3475,3479,3484,3492,3500,3508,3516],[3690,3693,3700,3708,3716,3726,3737,3752,3758,3767,3778,3783,3793,3800,3809,3816,3826,3837,3852],[4240,4250,4283,4292,4300,4308,4317,4327,4334,4342,4350,4353,4358],[4518,4532,4550,4559,4567,4575,4583,4596,4600,4614,4617,4628,4634,4647],[4853,4866,4875,4883,4893,4900,4901,4914,4917],[4964,4968],[5174,5188,5193,5201,5209,5217,5225,5234,5243,5251,5259,5264,5267,5278,5284,5296,5300,5312,5317,5328,5334,5342],[5744,5749,5751,5761,5767,5775,5784,5792,5801,5810,5817,5826,5834],[6085,6101,6106,6109,6117,6128,6134,6146,6151,6163,6168,6179,6184,6196,6201,6212,6217,6218,6230,6234,6246,6251,6263,6267],[6442,6456,6467,6470,6479,6484,6494,6501,6511,6518,6531,6534],[6712,6712,6720,6727,6734,6749,6751,6761,6768,6780,6784,6797,6801,6813,6818,6818],[7091,7105,7110,7118,7118,7126,7135,7144,7151,7161,7168,7182,7185,7196,7201],[7336,7394,7402,7410,7418,7430,7435,7446,7451,7463,7468],[7668,7684,7688,7693,7703,7710,7718,7728,7735,7745,7751,7759],[7778,7782,7790],[7978,7993,8002,8011,8018,8028,8035,8049,8051,8062,8068,8080,8085,8099,8101,8112,8118,8131],[8359,8372,8377,8385,8395,8402,8412,8418,8431,8435,8446,8451,8465,8468,8479,8485,8497,8502],[8667,8693,8716,8719,8719,8728,8735,8745,8752,8762,8768,8777,8785,8802,8813,8818],[8951,8965,8970,8985,8993,9002,9012,9018,9031,9036,9047,9052,9064,9068,9069,9078,9086,9100,9102,9113,9119,9129,9135,9152,9162,9169,9184,9185,9196,9202,9215,9219,9229,9235,9246,9247,9252,9263,9269,9280,9285,9298,9302,9314,9319,9330],[9483,9497,9503,9512,9519,9519,9528,9535,9544,9552,9565,9569,9581,9586,9598,9603]],"version":"2.0.0"}

    PAB=16+36=23

    Así, la probabilidad de obtener 1 en el primer lanzamiento y un número par en el segundo es 23{"x":[[314,314,314,314,314,317,319,327,334,341,344,354,356,359,361,363,363,361,356,350,343,337,331,329,324,323,322,323,325,329,334,337,345,359,365,381,391,400,410,418,425,428],[304,307,310,326,340,359,372,378,384,400,405,409,417,424,428,431,432],[363,360,360,359,359,360,363,367,372,375,383,386,391,393,393,393,392,388,386,382,380,379,377,376,376,380,384,388,392,397,402,402,402,399,397,389,385,373,363,354,350,347,344,342]],"y":[[186,185,183,181,180,176,174,170,168,166,166,166,167,169,170,178,181,192,200,209,218,227,234,238,245,247,251,253,253,254,254,254,252,249,247,243,241,239,237,236,235,235],[322,322,321,318,316,313,312,311,311,310,309,309,310,311,312,313,314],[388,388,387,387,386,384,381,378,376,375,372,371,371,372,375,378,379,386,389,396,398,401,405,408,409,413,415,418,421,424,430,434,436,442,444,450,453,458,460,460,458,456,452,448]],"t":[[0,6,16,21,29,39,46,56,62,72,79,89,96,96,104,112,123,129,139,146,156,162,173,179,189,196,206,214,227,229,239,246,256,262,273,279,289,300,306,312,322,329],[569,604,612,622,629,649,654,655,662,671,679,680,689,696,705,712,722],[955,969,972,979,979,988,996,1004,1013,1024,1029,1042,1046,1059,1062,1075,1079,1091,1096,1108,1113,1113,1125,1129,1141,1146,1157,1163,1175,1179,1196,1204,1214,1227,1229,1239,1246,1256,1263,1271,1279,1280,1289,1292]],"version":"2.0.0"}.

    Probabilidad condicional

    Consideramos dos sucesos, A y B, tales que la probabilidad del suceso A depende del suceso B. En otras palabras, el resultado de A será distinto dependiendo de si B ha ocurrido o no. Escribimos un suceso "A|B", que se lee como "suceso A dado que ha ocurrido el suceso B".

    La probabilidad que rige este tipo de sucesos se conoce como Probabilidad Condicional, en la que un suceso se basa en la condición de que haya ocurrido otro suceso.

    Regla del producto

    Utilizando el concepto de probabilidad condicional, podemos esbozar una fórmula para la regla del Producto . Se da de la siguiente manera,

    PAB=PB|A PA

    Verbalmente, 'la probabilidad de que ocurran A y B es igual al producto de la probabilidad de que B ocurra A y la probabilidad de que ocurra A'. También podemos extender esta ley a tres sucesos, a saber, A, B y C,

    PABC=PC|AB PB|A PA

    Se pueden derivar expresiones similares para tantos sucesos como se desee.

    Considera el siguiente ejemplo para comprender cómo la ocurrencia de un suceso depende de la ocurrencia de un suceso precedente.

    Consideramos 2 bolsas, una bolsa naranja y una bolsa negra. Hay 4 caramelos en la bolsa naranja y 5 caramelos en la bolsa negra. También hay 2 bombones en la bolsa naranja y 3 bombones en la bolsa negra. Se elige al azar un caramelo de cualquiera de las bolsas, ¿qué probabilidad hay de que el caramelo elegido sea un bombón?

    Solución

    Sea A el suceso de que el caramelo elegido sea un caramelo y sea B el suceso de que el caramelo se haya elegido de la bolsa naranja. Sea C el suceso de que la bolsa elegida fuera la negra.

    Aquí podemos ver que la probabilidad de obtener un caramelo depende de la bolsa elegida. Si el caramelo se elige de la bolsa negra, la probabilidad es diferente si se eligió de la bolsa naranja.

    Reglas de probabilidad, El diagrama de árbol de todos los sucesos, StudySmarter El diagrama de árbol de todos los sucesos, StudySmarter Originals

    Considera el diagrama anterior, aquí se ramifican todos los posibles sucesos para comprender mejor las probabilidades.

    (i) Si el caramelo se eligió de la bolsa negra, decimos que la "probabilidad de obtener el caramelo dado que procedía de la caja naranja".

    Según los sucesos que hemos definido, la probabilidad de este suceso se denota como PA|B{"x":[[163,163,163,163,163,164,164,165,166,166,167,167,168,168,168,168,168,169,169,169,170,170,171,171],[148,147,147,148,150,157,162,167,184,194,204,213,218,230,234,243,247,250,251,251,249,243,240,232,222,212,209,206,203,198,195,194],[341,339,329,327,324,321,315,311,308,300,292,285,282,277,275,269,268,267,268,274,277,287,290,295,303,308,312,322,331],[395,393,391,391,390,390,392,394,395,399,401,406,408,414,419,424,428,429,433,437,440,443,445,447,450,453,456,457,459,461,462,465,465,466,467,468,468,470,470,471,472,473,475,477,479,481,483,484,487,488,490,490,491],[388,390,392,402,411,417,425,432,439,445,453,458,462,463,466],[562,563,563,564,564,565,566,567,567,568,569,569,570,570,571,571,572,573,574,575,577,579,580,582,583,585,585,588],[662,663,663,664,664,665,665,665,664,664,663,662,662,661,661,662,663,664,667,669,672,674,675],[671,671,674,677,679,681,689,692,703,707,710,717,720,722,726,728,728,729,729,726,722,716,709,701,693,689,680,677,674,673,674,675,677,679,684,690,698,702,706,714,719,727,735,742,745,750,751,751,750,744,741,730,721,712,706,702,698,689,681,674],[788,790,800,803,806,812,818,827,832,835,836,837,837,835,832,829,827,820,816,808,800,792,788,779,776,771]],"y":[[209,211,213,216,219,233,238,257,272,286,302,311,338,348,378,439,445,458,470,473,477,480,485,487],[229,226,224,217,215,209,206,204,197,195,195,197,199,205,207,217,224,232,236,240,249,259,265,275,286,296,300,303,305,309,310,311],[226,226,226,229,231,235,244,250,256,270,285,300,309,324,333,358,366,390,398,415,420,431,434,437,440,441,441,441,438],[407,407,405,404,398,391,382,372,367,349,343,325,319,300,289,277,267,262,253,245,239,234,231,230,228,228,228,229,231,235,238,249,254,268,279,292,298,317,324,331,337,349,366,376,384,392,398,401,408,409,413,414,415],[343,341,341,339,338,337,337,336,335,335,334,334,334,334,334],[222,221,219,220,222,227,234,243,254,266,278,285,306,312,326,339,347,359,372,383,392,401,405,415,418,425,426,429],[239,239,242,244,247,261,267,283,289,295,302,314,327,340,358,370,379,383,396,399,408,410,412],[250,247,232,227,225,224,219,218,214,213,213,214,215,217,221,227,230,236,243,254,262,270,278,287,295,298,309,312,319,322,326,327,328,329,331,332,333,333,334,335,336,338,340,344,345,353,355,364,368,377,380,389,394,398,400,401,401,402,402,402],[213,213,213,216,219,226,234,248,258,269,274,288,303,318,334,343,350,366,374,391,404,418,423,438,441,448]],"t":[[0,8,14,21,30,36,44,53,63,69,80,86,96,103,113,140,144,153,163,169,170,180,186,194],[451,455,463,469,480,486,487,496,503,513,520,529,536,546,553,563,570,579,586,587,596,603,613,619,629,636,637,646,648,654,661,663],[1036,1045,1048,1053,1055,1062,1070,1070,1079,1086,1096,1103,1103,1113,1120,1133,1136,1146,1154,1164,1170,1180,1186,1187,1196,1203,1203,1216,1220],[1512,1526,1536,1545,1553,1563,1570,1580,1587,1596,1603,1615,1626,1630,1636,1646,1653,1663,1670,1680,1687,1700,1703,1704,1712,1720,1732,1737,1738,1746,1754,1763,1770,1780,1787,1795,1803,1813,1820,1820,1831,1837,1847,1853,1864,1870,1880,1887,1901,1903,1915,1920,1931],[2201,2215,2221,2230,2237,2247,2253,2263,2270,2280,2287,2297,2303,2314,2320],[2627,2637,2639,2641,2645,2654,2662,2670,2680,2687,2697,2704,2720,2721,2730,2737,2737,2747,2755,2766,2771,2782,2787,2796,2804,2815,2820,2830],[3090,3105,3108,3113,3120,3130,3137,3147,3154,3154,3164,3170,3181,3187,3197,3205,3214,3221,3230,3237,3250,3254,3258],[3509,3521,3524,3525,3529,3537,3547,3555,3564,3571,3571,3581,3587,3588,3598,3604,3613,3621,3630,3637,3650,3654,3666,3671,3683,3687,3700,3704,3718,3721,3731,3737,3738,3748,3754,3764,3771,3774,3781,3787,3788,3798,3805,3814,3821,3831,3838,3847,3854,3866,3871,3883,3887,3898,3904,3905,3914,3921,3930,3937],[4266,4275,4278,4281,4288,4298,4304,4315,4321,4331,4337,4348,4357,4366,4371,4371,4382,4388,4388,4398,4405,4413,4421,4431,4438,4448]],"version":"2.0.0"} y se lee como 'ha ocurrido A dado B'.

    (ii) Si el caramelo se eligió de la caja naranja, la probabilidad de obtener un caramelo se denota como PA|C y se lee como 'Ha ocurrido A dado C'.

    Volvamos al ejemplo que vimos antes, y calculemos la probabilidad utilizando la regla del Producto.

    Hay 2 bolsas, una naranja y otra negra. Hay 4 caramelos en la bolsa naranja y 5 caramelos en la bolsa negra. También hay 2 bombones en la bolsa naranja y 3 bombones en la bolsa negra. Se elige un caramelo al azar, halla la probabilidad de que el caramelo elegido sea un bombón y proceda de la bolsa negra.

    Solución

    Sea A el suceso de que el caramelo elegido sea un caramelo y sea B el suceso de que el caramelo se haya elegido de la bolsa naranja. Sea C el suceso de que la bolsa elegida era la negra.

    Reglas de probabilidad, El diagrama de árbol que indica las probabilidades condicionales relevantes, StudySmarterEl diagrama de árbol que indica las probabilidades condicionales relevantes, StudySmarter Originals

    Queremos hallar la probabilidad de que el caramelo elegido sea un caramelo dado que la bolsa es negra, por lo que queremos hallar PAC.

    Utilizando la regla del producto tenemos

    PAC=PA|CPC{"x":[[30,30,30,30,30,30,30,30,30,30,30,30,30,30,29,29,29,29,29,29,29,29,29,29],[25,25,25,26,29,35,41,46,52,54,60,62,65,66,68,69,69,69,64,62,54,51,43,40,38,35,30,28],[142,141,134,132,130,126,117,114,108,101,96,94,89,85,82,81,80,80,80,82,85,88,90,92,96,99,101,106,108,111],[147,146,145,144,143,142,141,141,141,141,143,145,148,150,157,159,166,171,175,177,179,182,185,187,189,190,191,193,194,196,196,197,198,199,199,199,199,199,199,199,199,199,198,198,199,199,199,200,200,200,201],[153,152,150,150,152,153,155,159,161,164,169,171,177,182,186,190,193,195,198,199,201],[246,245,243,242,242,240,240,239,239,239,241,242,246,250,254,256,261,263,268,270,277,279,285,287,292,295,297,298,299,300,301,301,302,302,302,302,302,302,302,302,302,302,302,303,304],[373,372,368,365,364,362,360,356,354,346,342,340,336,334,332,332,333,336,338,344,346,355,358,366,372,375],[394,395,397,398,401,407,410,416,417,420,421,421,420,417,414,410,405,404,400,398,397,395,392,392],[448,449,450,454,456,463,465,467,472,475,479,480,481,482,483,484,484],[443,443,441,440,439,438,437,435,436,439,441,448,450,460,463,474,477,486,488,489,490,491],[543,543,543,543,542,541,540,540,539,538,538,536,535,534,534,534,534,534,535,535,536,537],[539,538,537,537,537,539,541,545,550,552,560,562,569,571,575,576,576,577,577,576,574,573,571,568,566,564,560,557],[634,633,631,628,623,620,613,606,601,599,597,593,591,591,591,592,596,598,605,612,617,623,626],[653,653,652,652,652,652,654,655,656,658,660,661,664,666,669,671,673,676,678,682,683,686,687,689,690,691,692,692,693,694,695,696,697,698,699,699,699,700,700,700,700,702,703,703,704,704,705,705,705,705],[669,665,663,662,662,662,663,664,667,669,676,678,680,685,687,690,694,696,698,699],[737,737,739,739,740,740,740,739,739,738,735,734,733,732,731,731,731,732,733,734,734,735],[801,799,796,793,789,784,778,776,769,767,763,762,761,761,762,763,766,767,769,775,777,783,790,795,797],[820,820,825,827,830,832,837,840,842,842,842,842,842,841,840,837,836,834,831,830,826,822,819,817,813],[896,896,896,896,896,896,896,895,894,893,892,891,891,889,888,887,886,886,886,886,887,888,889,891,892,894],[896,895,894,893,893,893,894,896,897,900,902,907,911,915,917,920,921,921,921,922,922,921,919,917,914,910,906,902,899],[987,986,984,983,977,971,964,957,954,948,946,943,939,937,936,935,936,938,942,944,953,956,960,966,969,973],[1032,1031,1029,1028,1026,1022,1020,1015,1009,1003,1001,996,993,989,985,984,983,983,985,989,994,997,1008,1012,1023,1027],[1063,1064,1068,1069,1072,1073,1078,1080,1081,1084,1086,1086,1086,1086,1085,1083,1081,1079,1074,1071,1068,1060,1054,1048]],"y":[[231,233,236,243,248,264,270,287,298,310,322,328,346,352,362,371,380,384,390,395,398,399,400,399],[246,245,244,242,239,235,232,231,231,232,236,238,242,245,250,256,261,264,275,278,289,292,301,304,306,307,309,309],[236,236,235,235,235,237,245,248,257,267,277,282,293,305,317,323,341,346,357,366,372,377,379,381,383,384,384,384,384,383],[354,354,354,353,350,349,344,342,337,334,326,319,311,307,295,291,281,276,271,268,267,264,262,261,261,261,261,264,267,272,275,282,289,297,302,306,315,319,323,332,336,348,354,359,363,365,368,369,370,368,366],[319,319,319,318,317,316,315,314,313,312,311,310,309,308,307,307,307,307,307,307,307],[344,344,342,341,339,335,334,329,325,316,309,306,295,287,281,277,271,269,265,263,258,258,257,257,259,262,266,269,271,277,281,287,294,299,302,310,322,329,335,342,346,348,353,354,355],[272,272,271,271,271,272,274,277,279,288,295,299,306,313,318,326,331,334,335,338,338,338,338,336,334,332],[236,236,237,238,241,248,251,263,267,279,283,297,301,316,325,335,344,348,355,359,362,368,373,374],[284,284,284,283,283,282,282,282,281,281,281,281,281,281,281,282,283],[312,313,314,315,315,315,316,317,317,316,315,313,312,309,308,306,305,303,303,303,303,303],[256,257,260,263,276,281,297,302,308,318,324,340,345,354,361,371,376,377,380,381,382,382],[263,262,259,258,257,255,254,253,252,252,254,255,260,262,269,272,274,279,284,286,291,293,296,300,303,304,308,311],[254,254,254,256,260,264,273,285,295,301,307,323,333,338,352,356,364,367,371,373,373,373,372],[341,340,336,335,329,327,321,317,313,306,301,297,288,284,277,274,270,266,263,258,257,256,256,258,261,264,266,269,271,277,280,283,289,300,307,315,319,329,332,341,343,348,350,352,352,353,352,349,348,345],[307,308,308,309,310,311,311,311,311,310,308,307,306,304,303,302,301,300,300,299],[269,268,265,264,264,266,268,275,278,286,300,310,320,324,334,338,344,345,350,352,353,353],[275,275,275,275,277,281,287,290,301,306,313,319,322,327,330,331,333,334,334,335,335,335,334,333,332],[241,242,244,245,250,252,261,268,276,280,285,289,299,304,309,318,322,327,335,338,345,351,355,357,362],[261,265,267,274,278,283,294,299,305,315,321,326,331,342,346,351,362,367,372,375,376,377,377,375,373,369],[280,279,275,273,272,270,269,268,267,266,265,265,267,269,273,278,280,283,285,290,292,295,299,301,305,308,311,313,314],[282,282,282,282,283,286,291,298,301,309,314,318,327,331,338,346,351,355,358,360,362,363,363,363,363,363],[299,299,299,299,299,300,300,302,304,308,310,315,318,324,331,334,340,343,347,350,352,353,354,354,354,353],[271,272,274,275,279,282,290,293,296,302,309,312,320,324,328,337,341,345,353,356,360,369,374,378]],"t":[[0,40,47,57,64,74,81,91,97,107,114,124,131,141,147,158,164,164,174,181,191,197,208,222],[521,526,532,541,547,557,564,574,581,589,597,607,614,614,624,631,640,648,657,664,673,681,690,697,698,707,714,723],[1136,1152,1156,1157,1165,1173,1181,1191,1198,1208,1214,1215,1223,1231,1240,1248,1257,1265,1276,1281,1291,1297,1298,1308,1314,1315,1327,1331,1331,1344],[1600,1605,1608,1614,1631,1640,1648,1660,1664,1665,1676,1681,1696,1698,1710,1715,1725,1731,1741,1748,1748,1758,1764,1776,1781,1782,1791,1798,1810,1814,1815,1826,1831,1844,1848,1850,1860,1865,1867,1875,1881,1891,1898,1907,1915,1923,1931,1941,1948,1965,1974],[2176,2190,2193,2198,2208,2215,2223,2231,2232,2241,2248,2248,2258,2265,2275,2282,2293,2298,2311,2315,2327],[2751,2755,2758,2759,2765,2775,2782,2792,2798,2808,2815,2825,2832,2841,2848,2859,2865,2865,2875,2882,2891,2898,2908,2916,2927,2932,2942,2948,2949,2959,2965,2975,2982,2982,2992,2998,3008,3015,3025,3032,3042,3049,3058,3065,3077],[3500,3516,3520,3521,3525,3532,3532,3542,3548,3559,3565,3575,3582,3592,3598,3612,3616,3627,3632,3642,3649,3660,3665,3678,3682,3690],[3988,4025,4032,4042,4049,4059,4065,4076,4082,4092,4100,4109,4117,4126,4132,4142,4149,4159,4165,4166,4176,4182,4192,4200],[5102,5143,5150,5164,5166,5176,5182,5183,5191,5199,5209,5216,5216,5228,5232,5233,5249],[5505,5510,5513,5517,5524,5527,5534,5543,5567,5581,5583,5594,5599,5610,5617,5627,5633,5641,5649,5650,5660,5662],[5984,6016,6025,6033,6043,6050,6060,6066,6066,6077,6083,6093,6099,6110,6116,6127,6133,6143,6149,6150,6160,6167],[6414,6430,6434,6435,6441,6450,6460,6466,6477,6483,6493,6500,6512,6517,6525,6533,6533,6541,6550,6560,6566,6567,6575,6583,6583,6594,6600,6613],[6960,6968,6983,6993,7000,7010,7016,7027,7033,7034,7044,7050,7061,7067,7077,7083,7094,7101,7108,7116,7126,7133,7144],[7423,7435,7438,7443,7450,7461,7467,7467,7478,7483,7484,7496,7500,7500,7511,7517,7518,7528,7534,7543,7550,7560,7567,7581,7583,7594,7600,7601,7611,7617,7617,7627,7633,7644,7650,7661,7668,7677,7685,7697,7700,7711,7717,7727,7733,7743,7750,7767,7768,7777],[7929,7939,7946,7950,7967,7976,7983,7986,7995,8000,8011,8017,8017,8028,8033,8034,8042,8050,8051,8061],[8306,8320,8323,8328,8342,8350,8361,8367,8378,8384,8394,8402,8411,8417,8428,8434,8444,8452,8461,8467,8478,8484],[8832,8838,8843,8851,8861,8867,8878,8884,8894,8901,8911,8917,8918,8927,8934,8948,8951,8951,8959,8967,8978,8984,8995,9001,9009],[9323,9331,9334,9339,9342,9351,9362,9367,9376,9384,9385,9395,9401,9401,9412,9417,9418,9429,9434,9434,9445,9451,9461,9467,9478],[9775,9780,9785,9795,9801,9801,9812,9818,9818,9829,9834,9835,9846,9851,9851,9862,9868,9878,9884,9895,9902,9912,9917,9929,9935,9948],[10140,10154,10157,10162,10170,10178,10185,10195,10201,10212,10218,10229,10234,10246,10251,10262,10268,10268,10279,10284,10285,10296,10301,10301,10309,10319,10328,10335,10346],[10676,10691,10694,10701,10710,10718,10726,10734,10745,10751,10752,10763,10768,10768,10780,10785,10796,10801,10812,10818,10829,10835,10837,10846,10851,10855],[11134,11139,11142,11147,11152,11160,11160,11168,11176,11185,11195,11201,11202,11213,11218,11229,11235,11235,11247,11252,11263,11268,11276,11286,11295,11302],[11552,11564,11567,11570,11580,11585,11596,11601,11602,11613,11618,11629,11635,11635,11647,11652,11652,11664,11668,11669,11680,11685,11696,11702]],"version":"2.0.0"}

    La probabilidad de que el caramelo proceda de la bolsa negra es

    PAC=candies in the black bagtotal sweets in black bag=58

    y la probabilidad de elegir la bolsa negra es 1/2, ya que sólo hay dos bolsas,

    PC=12

    Sustituyendo estos valores, obtenemos

    PAC=58·12=516

    Por tanto, la probabilidad de que el caramelo sea un caramelo y proceda de la bolsa negra es 516{"x":[[534,534,533,532,526,524,521,515,512,509,507,501,496,491,490,488,485,483,480,477,475,472,471,471,471,472,472,473,474,475,476,477,478,478,478,477,477,477,477,478,478,479,481,482,483,486,488,490,492,496,498,500,505,508,512,517,521,523,528,530,533,534,534,534,534,531,527,522,517,514,508,502],[430,432,446,457,463,470,484,506,522,536,544,567,574,592,603,607,615,621,624,625],[450,450,450,450,450,449,447,445,444,441,439,436,435,431,430,430,429,429,429,429,429,430,432],[533,532,529,527,525,519,516,510,507,502,497,493,491,489,489,490,491,497,499,507,513,519,523,525,528,533,535,537,539,540,540,540,539,536,530,523,518,506,501]],"y":[[106,105,105,105,107,108,108,110,110,111,111,113,114,115,116,116,116,116,117,117,116,116,116,117,119,123,126,131,136,141,146,156,164,170,172,176,177,178,179,179,178,177,176,174,173,170,168,167,165,163,162,160,159,159,159,161,164,165,170,171,177,179,186,188,191,197,202,207,211,214,218,221],[278,278,275,274,273,273,271,268,266,264,263,261,261,260,260,260,261,262,263,264],[322,323,325,328,336,348,361,374,380,398,404,422,427,442,446,449,455,457,459,461,462,462,460],[315,314,311,311,312,320,324,336,343,357,373,388,403,418,425,441,445,451,452,453,450,446,443,441,437,430,426,422,416,413,410,406,405,404,405,408,410,417,419]],"t":[[0,23,32,39,49,56,56,64,72,73,83,89,97,106,106,116,122,123,133,140,150,166,189,199,206,216,223,233,240,247,256,282,292,303,306,314,323,333,339,350,356,356,366,376,377,383,389,390,400,406,406,416,423,423,433,439,450,456,466,474,483,489,500,506,507,517,522,532,539,549,556,566],[849,865,868,869,873,882,889,899,906,916,923,933,939,949,956,967,973,982,989,995],[1311,1311,1315,1316,1323,1333,1340,1350,1356,1366,1373,1383,1390,1400,1406,1407,1417,1423,1423,1433,1440,1452,1456],[1700,1716,1719,1723,1733,1740,1750,1756,1757,1766,1773,1785,1790,1801,1807,1816,1823,1833,1840,1849,1856,1866,1873,1874,1887,1890,1890,1901,1907,1907,1919,1923,1924,1934,1941,1952,1956,1966,1974]],"version":"2.0.0"}.

    Sucesos independientes

    Dos sucesos son Independientes entre sí si la ocurrencia de uno no afecta a la ocurrencia del otro de ninguna manera posible.

    Esto puede extenderse a cualquier número finito de sucesos, siempre que no afecten a la probabilidad del otro. Una propiedad importante de los sucesos independientes puede expresarse mediante una fórmula,

    PAB=PAPB{"x":[[81,81,80,80,80,79,78,78,77,76,76,76,76,76,76,77,77,77,77,78],[79,80,81,82,85,91,99,107,114,121,126,128,133,134,137,137,137,136,134,130,128,125,118,115,112,106,101,99],[220,219,217,216,215,211,208,203,198,192,187,184,175,172,165,161,158,158,158,160,162,168,170,177,180,189,192],[220,220,220,221,223,225,230,232,237,240,247,250,252,257,261,263,268,269,273,275,276,278,279,279,279,279,279,279,278,278,279,279,280,280,280],[232,232,234,237,240,245,248,253,256,258,262,265,266,271,272,274,275,276,277],[313,313,314,315,315,319,320,323,324,326,331,335,341,347,354,360,363,371,374,379,382,384,385,385,386,386,386,387,387,386,386,386,386,387,390,391],[449,450,451,452,452,452,450,449,446,445,444,443,443,443,443,444],[453,453,454,455,456,463,466,468,473,477,479,482,483,484,487,488,489,488,487,483,481,475,473,466,462,460,459,459,460,461,462,466,467,473,474,480,483,485,488,489,490,490,490,487,486,484,482,476,473,468,465,462,460,455,453],[512,512,512,512,513,515,519,521,530,533,536,538,546,549,552,553,555,555,556,555,552,550,544,541,533,528,524,523,520],[600,600,601,602,605,609,614,616,624,627,634,635,640,642,643],[600,601,602,603,604,608,610,615,618,625,628,635,637,643,645,647,649],[691,692,692,692,692,689,688,686,685,684,684,683,683,683],[694,695,700,702,704,712,715,721,723,725,725,725,723,722,717,715,709,707,705,701,698],[771,769,768,766,761,758,748,742,738,735,733,733,733,733,737,739],[782,783,786,787,788,789,791,794,797,800,804,806,811,813,815,817,819,819,820,821,821,821,821,821,822,822,823,824,824,825,825,825,826,826,827,828],[796,797,799,801,803,807,810,818,820,826,827],[858,859,861,862,863,864,867,868,871,874,874,874,874,872,869,866,863,860,858,854,853,851,851],[932,933,934,934,934,933,931,930,927,926,924,922,921,920,919,917,917,916,915,915,915],[943,944,949,952,955,959,961,962,965,966,968,968,966,964,959,954,949,945,941,939,938,936,934],[1013,1012,1010,1008,1007,1006,1004,1002,996,990,987,980,977,977,976,976,976,978,981,983,989,995],[1034,1036,1037,1039,1040,1040,1040,1039,1038,1037,1036,1036,1036,1036,1036],[1041,1041,1039,1039,1039,1039,1042,1043,1049,1053,1057,1059,1063,1064,1064,1064,1063,1060,1058,1055,1054,1053,1052,1051,1050,1050,1050,1050,1052,1053,1057,1059,1061,1065,1067,1069,1070,1070,1069,1067,1065,1064,1059,1057,1054,1049,1046,1044,1042,1038],[1094,1095,1099,1102,1102,1103,1103,1103,1103,1102,1101,1099,1097,1092,1089,1081,1077,1070]],"y":[[207,211,219,224,244,251,272,287,295,318,333,348,354,370,374,385,390,394,395,396],[216,215,213,212,210,208,206,205,205,206,209,211,218,220,228,234,241,244,248,255,259,263,271,274,276,281,284,285],[226,224,221,220,219,217,217,217,220,226,233,238,258,266,292,311,329,337,356,365,368,375,377,380,381,383,383],[345,343,337,330,323,319,305,300,286,280,266,261,257,248,241,238,233,232,230,230,231,237,243,251,261,272,283,288,304,313,320,324,332,336,337],[302,301,299,298,297,297,296,296,296,296,295,295,295,294,294,295,295,296,296],[336,334,327,325,321,311,307,298,294,290,276,267,257,247,239,233,230,227,227,230,235,241,249,257,266,276,286,295,305,315,323,330,333,338,340,339],[247,247,247,249,259,264,284,291,309,315,325,335,338,342,344,345],[257,255,243,242,241,237,235,234,233,232,232,232,232,233,235,236,239,242,243,250,253,263,267,278,285,291,295,296,297,298,298,300,300,303,304,310,315,317,325,327,333,335,337,339,341,342,343,346,347,349,350,351,351,351,351],[214,213,212,211,209,209,209,210,214,217,220,223,234,239,249,254,265,277,288,301,314,320,340,346,363,373,381,384,389],[275,274,274,274,275,275,276,276,276,276,276,276,276,276,276],[311,311,311,311,311,310,310,310,309,308,308,307,307,306,306,305,305],[237,236,237,244,248,264,270,288,293,304,312,316,320,330],[240,239,231,230,229,228,227,227,229,231,235,239,245,247,255,259,266,269,271,275,278],[248,248,250,252,260,265,285,298,311,323,334,337,341,344,350,351],[322,320,309,306,302,298,291,282,274,265,256,252,243,242,241,243,248,252,260,268,273,277,290,294,301,304,310,315,319,322,328,330,334,335,337,337],[287,286,286,286,286,285,284,282,281,280,279],[248,248,247,247,247,248,251,252,257,268,272,282,287,296,305,314,321,327,330,335,336,339,340],[245,245,246,248,251,265,272,278,291,297,303,314,319,325,329,337,341,345,354,359,361],[247,246,242,242,242,242,242,242,244,245,250,252,259,262,270,276,281,286,289,291,292,293,294],[257,257,254,254,254,255,256,258,266,275,279,294,303,307,315,319,327,332,335,336,336,335],[265,265,265,266,268,277,285,293,300,304,311,314,319,321,322],[284,283,275,274,272,271,268,266,262,260,259,258,259,260,262,267,269,276,279,283,285,286,288,289,290,291,292,293,295,296,301,304,306,311,313,317,318,321,322,324,325,326,327,328,330,332,333,334,335,336],[250,249,247,252,255,258,265,270,285,295,304,312,315,325,329,341,345,352]],"t":[[0,8,17,27,33,43,49,60,66,77,83,93,99,110,116,127,133,141,149,154],[457,462,465,466,477,483,493,500,510,516,526,533,543,550,561,566,578,583,583,593,600,600,610,616,617,626,633,643],[1191,1199,1202,1203,1208,1219,1229,1233,1243,1250,1260,1267,1277,1283,1293,1300,1310,1317,1326,1333,1343,1350,1360,1367,1378,1384,1392],[1685,1696,1700,1701,1710,1718,1727,1733,1743,1750,1760,1767,1767,1777,1783,1793,1800,1801,1810,1817,1826,1833,1846,1850,1862,1867,1876,1883,1893,1900,1913,1917,1931,1933,1944],[2201,2217,2220,2222,2225,2234,2246,2250,2251,2260,2267,2267,2277,2284,2294,2300,2311,2317,2325],[2667,2682,2685,2689,2692,2700,2711,2717,2717,2727,2734,2744,2750,2760,2767,2777,2784,2797,2801,2811,2817,2826,2834,2844,2850,2861,2867,2881,2884,2897,2901,2912,2917,2927,2951,2957],[3209,3228,3231,3237,3242,3252,3261,3267,3278,3284,3294,3301,3311,3317,3318,3328],[3539,3557,3560,3561,3568,3576,3584,3584,3594,3601,3611,3618,3618,3628,3634,3645,3652,3661,3667,3676,3684,3694,3701,3711,3717,3727,3734,3744,3751,3751,3761,3768,3777,3784,3794,3801,3812,3817,3830,3834,3846,3851,3851,3860,3868,3868,3878,3884,3885,3895,3901,3902,3912,3918,3922],[4286,4293,4294,4301,4311,4318,4328,4334,4345,4351,4351,4362,4368,4378,4384,4385,4395,4401,4411,4418,4429,4434,4445,4451,4463,4468,4478,4484,4493],[4947,4966,4976,4985,4995,5001,5012,5018,5028,5035,5045,5052,5062,5068,5079],[5261,5291,5297,5306,5309,5314,5322,5328,5335,5345,5351,5362,5368,5376,5385,5385,5395],[5610,5627,5631,5643,5652,5662,5668,5679,5685,5695,5702,5702,5712,5718],[5968,5982,5985,5989,5993,6002,6012,6018,6029,6036,6046,6052,6062,6068,6079,6085,6096,6102,6102,6113,6118],[6495,6512,6519,6520,6530,6535,6546,6552,6562,6569,6579,6585,6586,6594,6602,6610],[6894,6908,6911,6912,6919,6919,6929,6936,6946,6952,6963,6969,6979,6985,6996,7002,7013,7019,7031,7035,7036,7047,7052,7063,7069,7069,7082,7085,7099,7102,7113,7119,7129,7136,7146,7157],[7353,7386,7394,7402,7403,7413,7419,7430,7436,7446,7452],[7735,7753,7756,7761,7769,7769,7777,7786,7796,7802,7813,7819,7820,7830,7836,7847,7853,7864,7869,7881,7886,7897,7902],[8157,8171,8174,8179,8186,8197,8202,8203,8214,8219,8220,8231,8238,8239,8249,8252,8253,8264,8269,8280,8286],[8542,8557,8560,8562,8569,8578,8586,8589,8597,8603,8614,8620,8630,8636,8647,8653,8664,8669,8680,8686,8686,8698,8703],[8919,8937,8940,8945,8948,8953,8953,8963,8969,8980,8986,8997,9003,9014,9019,9020,9031,9036,9047,9053,9064,9070],[9287,9290,9295,9303,9314,9320,9331,9336,9347,9353,9364,9370,9381,9386,9389],[9541,9558,9562,9566,9570,9578,9586,9597,9603,9614,9620,9628,9637,9647,9653,9664,9670,9681,9686,9697,9703,9704,9715,9720,9720,9731,9736,9748,9753,9765,9770,9781,9787,9798,9803,9815,9820,9831,9836,9848,9853,9854,9865,9870,9871,9881,9886,9887,9898,9903],[10087,10105,10108,10111,10120,10122,10132,10137,10148,10153,10164,10170,10183,10187,10196,10203,10214,10220]],"version":"2.0.0"}

    En otras palabras, la probabilidad de la intersección de dos sucesos independientes es el producto de las probabilidades individuales.

    Jason y William están jugando a las cartas, Jason le pide a William que saque una carta al azar. William saca una reina y la devuelve a la baraja. Jason le pide que saque otra carta y le pregunta la probabilidad de que esta carta sea un rey seguido de la reina anterior. ¿Cuál debería ser la respuesta de Guillermo?

    Solución

    Sea A el suceso de que la carta sacada sea una reina y B que la segunda carta sacada sea un rey.

    Hay que tener en cuenta que no importa qué elige Guillermo como primera carta, ambos sucesos son completamente independientes entre sí.

    Calculando las probabilidades individuales, obtenemos

    PA=PB=452=113

    Como hay cuatro reinas y cuatro reyes en una baraja de 52 cartas, queremos hallar la probabilidad de la intersección de los dos sucesos, utilizando el hecho de que los sucesos son independientes.

    PAB=PA PB=113113=1169.

    ¿Son independientes los dos sucesos siguientes?

    A : La salida del sol

    B : Lanzar una moneda y que salga cara

    Solución

    ¡SÍ lo son!

    Los sucesos A y B son independientes porque no tienen ninguna relación entre sí y el hecho de que ocurra uno no afecta al otro. La salida del sol es, sin duda, independiente del resultado de lanzar una moneda.

    Complemento de un acontecimiento

    Supongamos que se lanza una moneda justa una vez, hay dos resultados posibles: Cara (H) o Cruz (T), con probabilidades iguales. Ambos sucesos son exactamente opuestos entre sí, si se produce un suceso, todos los demás resultados excepto éste se conocen como sucesos Complementarios del mismo. Si al lanzar la moneda obtenemos H, cualquier otro suceso (en este caso, T) es un suceso complementario de él.

    El complemento de un suceso, digamos A, es el subconjunto del espacio muestral que no está contenido en el propio A. Se denota porAc.

    Esto implica que la probabilidad del complemento vendrá dada por todos los elementos que estén en el espacio muestral excluyendo el conjunto A.

    AC=S-A{"x":[[109,131,138,146,148,152,155,160,165,170,172,177,183,186,188,189,192,193,196,197,197,198,198,198,197,197,195,194,193,192,192,191,191,192,193,194],[128,130,132,135,145,152,158,161,164,170,173,175,177,185,187,192,194,197,198],[274,273,271,268,263,261,255,252,245,243,241,236,232,230,229,229,230,232,236,239,241,248,259,267],[304,305,306,310,316,323,332,340,349,358,366,375,392,398,404,406],[316,317,321,326,333,337,350,358,366,370,374,377,385,387,390,396,398,402,404],[578,581,582,584,589,591,592,593,593,592,591,585,582,579,572,563,558,547,530,519,514,504,499,492,487,485,484,487,489,498,502,516,522,527,539,544,549,559,567,571,576,578,579,581,581,580,576,570,566,552,547,532,527,513,508,504,497,494],[642,643,646,651,654,661,665,669,679,688,697,702,710,718,725,728,735,737,740,741,741,740,737,736],[829,829,830,830,831,832,836,838,841,844,848,850,859,862,872,875,884,887,890,895,898,899,902,905,906,908,909,909,910,910,909,908,907,907,907,907,908,910,911,912,913,914,915,915,915,916,917,917,918],[862,862,864,866,871,876,883,891,898,905,908,915,917,920]],"y":[[347,271,254,237,233,224,220,212,206,201,198,195,193,194,195,197,203,207,221,233,239,260,267,288,294,307,326,338,350,356,370,374,384,389,392,393],[293,293,293,293,292,291,291,290,290,289,289,289,289,288,288,287,287,286,286],[134,133,131,130,131,132,137,140,149,153,157,166,177,186,193,197,202,206,209,210,211,211,211,209],[267,267,266,265,264,262,261,259,258,256,255,254,252,251,251,251],[321,321,321,320,318,318,315,313,311,311,310,309,308,307,307,306,306,306,306],[240,235,232,228,221,216,214,205,203,195,193,187,186,185,184,184,186,189,197,204,208,217,222,231,240,248,252,262,265,272,274,280,282,284,288,290,291,296,300,303,308,311,313,319,322,328,335,341,344,353,355,361,362,364,364,363,360,358],[284,284,284,285,285,285,285,285,284,283,282,282,281,281,281,281,281,281,282,282,284,284,286,287],[348,347,344,342,337,334,320,315,302,295,279,271,246,237,213,206,186,181,176,167,164,162,160,159,160,163,165,169,179,198,212,228,235,257,265,282,288,302,310,316,323,329,332,335,338,342,347,349,350],[284,285,286,287,288,289,289,288,287,285,285,284,283,283]],"t":[[0,12,17,33,36,44,44,54,61,71,77,88,94,104,111,111,121,127,137,144,154,161,171,177,188,194,204,211,221,227,238,244,254,261,271,278],[794,818,822,828,838,844,854,861,861,871,878,878,888,894,904,911,921,928,928],[1377,1386,1390,1391,1403,1412,1422,1429,1438,1445,1445,1459,1461,1472,1478,1488,1495,1505,1511,1512,1520,1528,1538,1545],[1938,1944,1953,1961,1971,1978,1988,1995,2005,2012,2023,2029,2053,2062,2072,2078],[2291,2328,2337,2345,2355,2362,2372,2379,2393,2395,2395,2409,2412,2412,2421,2428,2429,2439,2445],[2863,2872,2882,2889,2895,2906,2912,2922,2929,2937,2945,2955,2962,2962,2970,2978,2989,2995,3006,3013,3022,3029,3029,3039,3045,3055,3063,3072,3080,3089,3095,3105,3112,3113,3122,3129,3129,3139,3145,3156,3162,3163,3172,3179,3180,3189,3195,3205,3212,3222,3229,3237,3246,3255,3262,3263,3272,3279],[3829,3837,3854,3862,3872,3879,3879,3889,3896,3904,3912,3913,3923,3930,3939,3946,3956,3962,3972,3979,3989,3995,4006,4012],[6730,6735,6738,6740,6747,6756,6763,6772,6780,6781,6790,6797,6807,6813,6824,6833,6843,6847,6847,6857,6863,6864,6874,6880,6890,6897,6897,6907,6913,6924,6931,6940,6947,6957,6963,6974,6980,6991,6997,7008,7013,7024,7030,7031,7041,7047,7059,7064,7074],[7456,7475,7497,7505,7514,7524,7531,7541,7548,7557,7565,7574,7580,7591]],"version":"2.0.0"}

    PAC=PS-PA=1-P(A)

    que también se conoce como Regla del Complemento de la probabilidad.

    Un espacio muestral se define por el conjunto S = {1,2,5,7,8,9} y un subconjunto de S viene dado por

    A = {2,5,8}. Halla el complemento del conjunto A.

    Solución

    Según la definición de complemento de un conjunto, todos los elementos que están en S pero no en A forman el conjunto deAc, que aquí forman el conjunto {1,7,9}.

    Por tanto, el complemento de A esAc = {1,7,9}.

    Se lanza una moneda dos veces y se observa el resultado. ¿Cuál es la probabilidad de que se observe al menos 1 cara?

    Solución

    El espacio muestral viene dado por S = {HH,HT,TH,TT} y sea A el conjunto (o el suceso) formado por los elementos cuando tenemos más de 1 cara.

    El complemento de este suceso será "el suceso cuando tenemos menos de 1 cabeza". Por tanto, Ac vendrá dado por el conjunto {TT}, ya que es el único elemento del espacio muestral en el que tenemos menos de 1 cabeza (ninguna cabeza).

    Por tanto, ahora podemos utilizar la regla del complemento para calcular su probabilidad

    PA=1-PAC

    PA=1-14=34

    Por tanto, la probabilidad de obtener al menos 1 cabeza es 34{"x":[[464,465,466,470,472,478,485,495,498,505,509,513,516,519,519,518,517,511,508,498,491,485,482,476,471,470,469,471,475,479,485,490,495,498,505,507,510,511,511,507,505,494,489,474,469,454,446,442],[387,388,396,400,416,422,440,446,466,473,480,495,501,515,536,547,558,563,575,581,586,587],[486,485,478,477,476,473,471,468,467,464,461,458,457,456,456,459,463,468,475,483,488,502,512,516,525,530,538,545,551,556,558,559],[506,505,504,504,504,504,504,503,503,503,502,501,500,500,500,500,501]],"y":[[115,112,108,105,105,103,103,102,102,101,101,101,103,106,108,115,119,130,134,147,153,159,162,166,170,172,173,174,174,175,176,178,180,182,186,187,191,193,198,202,204,213,217,226,229,236,239,240],[291,289,288,287,286,285,283,282,280,279,278,277,276,275,274,274,274,274,274,275,276,277],[306,307,312,315,318,332,337,347,351,360,370,378,382,393,395,402,405,407,408,409,409,408,407,407,406,405,405,404,404,404,404,404],[362,361,363,369,378,384,390,404,415,421,438,448,456,464,469,472,473]],"t":[[0,5,12,18,28,34,45,51,61,68,68,76,84,95,101,111,118,128,134,145,151,161,168,178,184,195,201,218,228,234,245,251,261,268,278,284,295,301,309,318,328,334,345,351,364,371,378,384],[664,682,693,701,710,718,728,734,745,751,752,760,768,778,785,795,801,811,818,828,835,843],[1065,1081,1084,1085,1093,1101,1111,1118,1118,1128,1135,1145,1151,1165,1168,1178,1185,1193,1201,1210,1218,1228,1235,1245,1251,1252,1264,1268,1282,1285,1295,1301],[1469,1480,1494,1502,1511,1518,1518,1529,1535,1543,1551,1562,1570,1581,1585,1597,1602]],"version":"2.0.0"}.

    Reglas de probabilidad - Puntos clave

    • La unión y la intersección de cualquier número de sucesos se pueden relacionar mediante La regla de adición de la probabilidad y para 2 sucesos cualesquiera A y B, viene dada porPAB=PA+PB-P(AB). Para dos sucesos disjuntos, A y B, la regla de adición viene dada por PAB=PA+PB{"x":[[68,69,71,72,74,76,77],[71,70,69,68,68,67,68,71,73,81,84,95,99,109,115,117,121,123,124,124,122,118,104,97,93,82,80,74,73,72],[195,196,197,195,191,186,180,176,162,156,144,136,130,128,126,123,121,121,120,121,123,126,128,133,136,146,149,153],[189,187,184,184,183,183,183,183,185,187,190,192,199,202,210,212,218,220,221,224,228,231,233,234,236,237,238,239,239,239,239,240,241,243,243,245,246],[184,181,180,183,187,190,201,206,221,226,239,242,249,251,252,253],[301,301,300,300,300,300,300,302,302,304,306,309,313,317,319,324,326,329,334,341,343,349,350,355,356,357,358,357,356,356,353,352,351,350,347,346],[394,395,396,397,400,400,402,402,403,403,403,403,402,402,401,401,400,400,399,399,399,399],[390,389,389,390,394,400,403,413,417,426,432,435,437,439,439,438,436,433,429,423,419,416,414,413,413,416,417,423,425,431,433,435,439,441,442,443,445,445,444,444,442,439,433,427,425,422,415,408,405],[468,469,471,472,478,481,486,489,493,495,497,498,497,496,491,489,481,477,468,465,461,455,453,448,445],[534,532,532,533,535,538,543,549,552,555,560,567,568,570,570,569,568,567],[531,533,536,543,546,554,562,566,570,576,579,584,589,593,594,598,599,600],[636,637,638,639,639,640,640,639,638,637,637,636,635,635,635,635,635,636,636,637,638,639],[631,631,634,635,638,640,643,646,651,657,664,668,671,675,675,672,671,664,661,659,654,649,645],[724,727,728,727,724,722,717,707,704,698,696,693,693,694,697,699,704,709,716,724,731],[760,759,758,758,758,758,758,759,760,764,766,768,770,774,777,781,785,788,789,791,793,796,797,800,801,804,805,805,806,806,806,806,806,806,806,805,805,804,804],[762,759,759,758,758,759,760,765,767,774,777,786,789,797,800,803,807,811,813,815,816,817,818],[837,841,845,848,851,853,854,858,859,860,860,859,857,855,851,849,846,840,838,835,831,829,827,823,821],[886,888,893,895,898,905,908,912,919,923,926,932,935,938,941,942],[910,912,913,914,915,915,917,917,918,918,918,918,918,918,918,918,918],[972,972,973,973,972,971,970,969,969,968,967,966,966,966,966,967,967,967,967,968],[968,970,973,977,979,981,989,991,994,998,1001,1003,1003,1003,1001,998,994,987,982,979,977],[1046,1045,1042,1041,1040,1037,1033,1030,1028,1027,1026,1025,1024,1024,1025,1027,1032,1036,1041,1043],[1065,1064,1063,1063,1062,1062,1061,1060,1060,1059,1058,1057,1057,1056,1056,1056],[1064,1064,1064,1065,1067,1071,1073,1079,1081,1083,1087,1090,1091,1092,1092,1090,1088,1085,1080,1076,1072,1070,1067,1067,1067,1068,1072,1073,1079,1083,1087,1090,1091,1093,1094,1094,1095,1095,1094,1093,1090,1087,1086,1079,1077,1072],[1119,1123,1124,1126,1127,1128,1129,1129,1130,1131,1131,1130,1127,1125,1116,1113,1102,1098,1087,1084]],"y":[[176,176,372,374,376,376,375],[181,181,180,180,179,178,178,174,173,169,168,166,166,167,170,172,177,180,186,193,202,210,228,236,240,249,250,253,253,253],[202,200,199,200,202,206,212,215,231,239,259,273,289,296,304,319,327,333,339,350,360,368,371,376,377,378,378,377],[328,325,319,317,308,304,295,290,279,269,258,253,237,233,221,218,212,211,210,211,213,218,224,227,236,250,260,271,282,293,298,311,315,325,327,331,331],[271,271,271,270,269,269,266,264,259,257,253,252,250,250,250,249],[241,244,251,255,259,267,271,284,288,292,301,309,313,315,316,316,315,314,311,302,299,286,281,268,263,249,245,234,231,228,223,221,218,216,211,209],[226,226,227,228,236,240,254,259,273,278,282,291,295,299,307,311,314,318,323,327,329,325],[238,231,229,224,219,214,212,206,204,202,202,204,205,210,215,218,224,230,236,247,253,258,264,268,270,276,277,283,284,290,292,294,298,299,301,303,306,310,314,316,318,322,328,332,333,335,337,338,338],[181,181,183,185,194,199,210,216,227,238,250,257,274,280,298,303,320,326,339,343,347,353,355,358,359],[240,240,239,239,239,239,238,238,238,237,237,237,237,238,239,241,243,244],[289,288,287,284,283,281,279,279,278,277,277,276,275,274,273,272,272,272],[196,194,196,199,206,211,224,239,247,262,269,285,291,309,314,326,329,337,339,340,341,341],[218,209,201,199,198,197,196,195,194,194,196,199,201,208,211,219,222,231,233,236,241,245,248],[206,204,203,206,210,213,220,238,244,266,273,292,298,314,322,325,331,334,335,335,334],[320,315,312,308,300,293,284,274,269,252,247,242,238,229,225,220,217,216,216,219,224,235,240,254,259,273,277,283,287,296,300,304,312,315,318,324,327,329,330],[282,280,279,278,277,277,277,276,275,273,273,271,271,269,269,268,267,267,267,266,266,266,267],[200,198,202,205,212,217,222,239,251,264,277,288,301,306,316,321,326,336,340,344,350,353,354,357,358],[268,268,266,266,265,265,264,264,263,263,263,263,263,263,262,262],[238,237,237,239,241,243,252,256,260,268,272,276,280,287,290,293,297],[206,207,208,210,213,233,240,262,269,290,303,315,324,328,336,340,342,347,348,351],[224,217,213,211,210,209,208,208,209,211,214,218,223,225,231,236,242,250,254,258,259],[232,232,233,234,236,242,251,262,273,279,285,301,306,319,322,327,331,332,333,332],[238,239,243,251,256,261,266,280,289,298,305,311,316,318,320,321],[251,243,239,235,233,227,225,221,221,220,220,220,222,225,227,233,236,241,249,254,259,262,268,269,273,274,277,278,282,285,289,292,294,298,300,301,305,309,312,315,318,320,321,323,323,323],[203,206,211,220,236,241,252,258,271,284,296,308,321,326,344,349,364,368,378,380]],"t":[[0,2,9,9,20,25,32],[439,443,449,451,458,469,474,485,492,502,509,522,524,535,541,551,558,558,568,575,585,592,608,618,625,635,642,651,658,666],[872,888,891,908,918,926,935,941,952,958,968,975,987,991,992,1000,1008,1008,1017,1025,1035,1041,1042,1051,1058,1068,1075,1079],[1294,1310,1313,1318,1325,1335,1342,1342,1352,1358,1369,1375,1387,1392,1402,1409,1418,1425,1425,1437,1442,1452,1459,1469,1475,1483,1492,1502,1512,1521,1525,1535,1541,1552,1559,1569,1575],[1769,1785,1788,1793,1802,1808,1819,1825,1836,1843,1852,1859,1869,1875,1875,1886],[2120,2138,2141,2143,2147,2150,2158,2169,2175,2175,2186,2192,2202,2208,2219,2225,2226,2236,2242,2252,2259,2271,2275,2286,2292,2304,2309,2317,2325,2326,2336,2342,2342,2352,2358,2371],[2544,2559,2568,2575,2586,2593,2603,2609,2619,2625,2625,2636,2642,2642,2653,2659,2659,2667,2675,2686,2692,2709],[2836,2846,2849,2850,2859,2869,2876,2886,2893,2902,2909,2919,2925,2936,2942,2953,2959,2969,2975,2986,2992,3003,3010,3019,3026,3035,3042,3052,3059,3070,3075,3076,3088,3092,3093,3103,3109,3121,3126,3126,3137,3142,3153,3159,3168,3174,3177,3186,3193],[3378,3394,3397,3402,3415,3420,3426,3426,3434,3442,3453,3459,3470,3476,3486,3492,3503,3513,3522,3526,3526,3536,3542,3553,3559],[3810,3825,3829,3842,3853,3859,3867,3876,3876,3886,3892,3903,3910,3920,3926,3936,3942,3946],[4091,4118,4126,4136,4142,4151,4159,4160,4170,4176,4176,4187,4193,4204,4209,4220,4227,4234],[4428,4445,4448,4452,4459,4460,4471,4477,4487,4493,4493,4504,4513,4523,4526,4536,4544,4554,4559,4560,4571,4576],[4731,4748,4752,4753,4759,4760,4768,4768,4776,4787,4794,4803,4811,4820,4826,4837,4843,4854,4859,4860,4871,4876,4887],[5053,5069,5073,5085,5093,5103,5110,5121,5127,5137,5144,5154,5160,5171,5176,5187,5193,5204,5210,5221,5226],[5411,5427,5431,5435,5443,5454,5460,5471,5476,5487,5493,5493,5504,5510,5510,5525,5527,5535,5543,5554,5560,5571,5577,5587,5593,5604,5610,5610,5621,5626,5627,5638,5643,5644,5655,5660,5660,5668,5677],[5807,5818,5822,5827,5830,5838,5843,5854,5860,5871,5877,5888,5894,5905,5910,5910,5921,5927,5938,5943,5943,5955,5960],[6137,6155,6159,6160,6168,6169,6177,6187,6193,6204,6210,6221,6227,6238,6243,6244,6255,6260,6260,6272,6277,6277,6288,6293,6296],[6507,6525,6528,6535,6536,6543,6544,6554,6560,6560,6572,6577,6577,6588,6594,6605],[6746,6757,6761,6766,6769,6777,6788,6793,6794,6805,6810,6811,6822,6827,6827,6839,6844],[7040,7050,7054,7055,7060,7071,7078,7088,7094,7103,7110,7121,7127,7138,7144,7144,7156,7160,7161,7172],[7345,7357,7361,7369,7369,7377,7389,7394,7394,7406,7410,7422,7427,7435,7444,7454,7460,7472,7477,7489,7494],[7666,7682,7686,7687,7694,7702,7711,7721,7727,7728,7739,7744,7756,7761,7772,7777,7788,7794,7805,7811],[7997,8015,8018,8020,8027,8028,8037,8044,8055,8061,8072,8077,8089,8094,8105,8111],[8233,8249,8253,8254,8261,8270,8278,8289,8294,8295,8304,8311,8322,8327,8339,8344,8352,8361,8372,8378,8389,8394,8405,8411,8422,8428,8439,8445,8452,8461,8471,8477,8489,8494,8495,8505,8513,8522,8531,8541,8545,8555,8561,8572,8578,8586],[8763,8777,8781,8782,8794,8805,8811,8811,8821,8828,8839,8844,8856,8861,8872,8878,8889,8894,8906,8911]],"version":"2.0.0"}.
    • Para dos sucesos cualesquiera, A y B, la regla del Producto de la probabilidad viene dada por PAB=PA|BPB{"x":[[28,28,28,28,29,29,29,29,28,27,26,26,26,26,26,26,27,27,28,28,30,30,31,32],[19,18,16,15,15,15,21,24,36,40,51,55,63,65,66,67,69,70,70,70,66,62,58,55,47,44,37,35,31],[109,104,99,97,94,92,87,84,73,69,67,67,66,67,70,74,78,80,83,88,90,93,96,102],[112,111,110,111,112,112,113,113,114,116,117,118,121,125,130,137,140,147,149,153,154,158,159,160,162,163,164,164,164,164,164,164,164,164,164,163,163,163,163],[116,112,110,111,115,118,123,129,136,140,144,150,159,161],[191,188,186,186,186,186,187,188,189,191,193,196,200,208,213,218,220,226,227,231,232,235,235,237,238,238,238,238,238,238,238,238,239,240,241],[281,280,280,280,280,281,282,282,282,282,281,280,280,279,279,279,279,280,281],[275,274,272,271,270,270,271,276,277,279,283,285,288,292,294,297,300,304,307,308,308,307,306,301,299,293,291,287,285,285,285,286,288,290,292,296,298,304,309,314,316,321,322,324,324,323,322,320,317,314,307,301,298,292],[337,339,341,345,347,349,355,358,360,361,362,363,363,363,363,362,361,358,355,348,345,338,335,327,325,318],[395,397,398,405,411,418,425,428,436,439,446,447,451,452,452],[393,396,398,407,413,420,426,428,433,435,439,440,441,442,442],[502,502,503,503,504,504,504,504,502,501,500,499,498,498,497,497,497,497,498],[502,501,501,503,504,505,509,514,515,521,523,528,529,531,531,531,530,527,522,520,517,514,508,506,503,497,493,491],[601,599,598,597,596,590,587,584,578,573,568,565,563,560,559,558,558,559,561,563,567,572,577,580,590,593,601],[625,625,625,626,628,629,632,634,635,637,640,641,642,646,647,651,655,658,659,661,662,664,664,667,668,668,669,670,671,671,671,671,671,671,670,670,669,669,668],[642,640,639,640,644,648,653,659,664,668,671,671],[700,701,702,703,704,704,704,704,703,703,702,702,701,700,700,699,699,698,698,698,698],[738,739,740,741,742,743,743,744,744,745,745,745,745,745,745,743,743,742,742],[746,745,744,745,746,748,750,753,754,758,760,761,765,768,768,769,769,769,767,766,763,762,758,755,752,751,751,750,750,750,751,753,755,758,762,767,771,775,777,782,783,785,785,785,785,784,783,780,778,776,772,769,767,763,761,759],[784,787,788,791,795,799,804,807,813,815,818,819,819,819,818,817,813,812,810,805,803,798,793,789,786],[870,870,870,870,870,870,870,870,870,869,869,869,869,868,868,868,868,868,869,869,870,870],[864,866,867,869,873,876,880,885,889,891,893,894,894,894,892,891,886,884,879,876,874,872,871],[941,940,937,936,933,931,929,924,919,916,914,914,912,912,912,912,912,912,914,917,918,921,922,924,928,930,932,937,939,943],[959,959,959,959,959,959,959,959,959,959,959,959,959,960],[957,957,958,959,960,962,967,969,972,976,979,980,982,982,982,982,980,978,977,975,973,972,969,967,966,963,963,963,964,967,970,972,977,982,986,990,993,995,996,999,1000,1000,1001,1001,1000,999,998,995,993,987,985,978,975,967],[1012,1013,1014,1016,1019,1023,1027,1029,1035,1037,1040,1041,1041,1039,1036,1035,1033,1030,1028,1026,1025,1022,1020,1017,1015]],"y":[[231,229,227,226,227,228,232,241,251,272,288,296,313,322,340,356,371,382,393,397,410,412,418,419],[236,234,232,229,225,222,215,212,205,205,205,207,212,214,217,219,225,228,231,238,249,257,265,269,281,284,294,296,301],[246,245,245,247,249,252,260,266,299,313,327,334,354,360,373,380,386,388,390,393,394,394,394,393],[361,357,353,344,341,337,334,325,317,307,301,296,286,275,267,258,256,253,253,258,260,270,274,278,286,291,305,315,320,331,336,340,349,353,357,363,366,370,372],[320,320,321,321,320,319,317,315,313,312,311,309,307,307],[355,355,353,351,347,344,340,333,329,319,308,296,283,268,261,254,252,249,249,252,254,264,268,281,291,295,308,313,321,333,340,342,346,348,349],[268,268,269,270,274,282,286,296,307,317,326,334,339,345,349,353,354,355,352],[277,276,274,270,269,263,261,254,252,250,247,246,244,243,243,243,244,247,250,254,260,266,270,280,285,295,298,306,309,311,313,314,315,315,316,316,316,317,319,321,322,326,327,332,333,338,340,342,347,349,354,358,359,361],[224,222,222,225,227,230,240,248,258,264,270,282,294,300,311,321,331,340,349,362,365,376,379,386,387,389],[286,285,285,284,283,282,282,282,282,282,282,282,283,284,285],[319,318,317,316,315,314,313,312,311,311,311,311,311,311,312],[241,242,245,252,263,269,281,294,311,323,329,338,346,353,360,362,365,366,366],[247,242,238,235,233,232,230,227,227,228,229,235,237,246,249,252,258,264,271,274,277,279,285,287,289,292,293,293],[231,230,230,230,231,237,240,245,256,268,282,289,297,311,319,333,346,356,365,369,374,378,381,382,383,382,381],[329,328,327,326,320,317,304,299,294,288,277,272,267,257,252,244,238,235,234,235,236,244,248,261,266,270,278,286,293,299,306,309,314,318,323,326,327,329,330],[285,285,285,285,284,284,282,280,279,277,277,276],[238,235,233,234,238,246,251,269,283,290,302,311,320,326,333,336,342,347,350,351,352],[251,252,252,256,264,274,286,291,296,305,314,318,324,327,329,335,338,339,340],[261,257,255,253,252,250,248,246,245,244,244,244,244,246,248,251,252,254,258,260,267,269,277,281,286,287,288,290,292,293,293,295,295,296,297,298,299,301,302,306,308,313,314,319,321,323,325,328,330,331,334,336,337,339,340,340],[221,221,221,223,226,232,240,244,258,263,278,283,288,293,307,312,321,326,330,338,342,350,356,361,365],[240,241,247,257,263,279,284,289,299,308,316,320,323,329,334,338,340,343,345,346,345,342],[244,241,240,238,236,236,235,235,236,237,241,245,249,251,258,259,265,266,271,273,275,277,278],[241,241,241,241,242,244,245,251,259,269,278,283,296,300,304,310,313,316,320,323,325,327,328,328,328,328,327,326,325,322],[255,262,273,281,290,294,303,306,314,315,317,320,321,321],[261,257,253,252,251,249,246,245,244,244,244,245,248,250,254,256,261,265,268,272,274,276,281,283,284,288,289,290,290,290,290,290,290,290,291,292,295,297,299,302,306,307,311,313,314,318,320,323,324,328,329,331,331,332],[233,233,232,231,233,236,241,245,258,263,277,282,296,305,314,319,322,330,334,337,341,347,350,355,358]],"t":[[0,7,12,14,30,41,47,58,63,74,81,91,97,97,108,114,124,131,141,147,158,163,174,180],[426,426,431,431,441,448,458,465,474,480,491,497,508,513,514,524,530,531,540,547,557,565,574,581,590,597,607,615,624],[1094,1110,1113,1115,1122,1123,1130,1141,1158,1165,1172,1180,1191,1197,1207,1214,1224,1230,1231,1241,1247,1248,1257,1264],[1499,1516,1519,1523,1530,1531,1541,1547,1558,1564,1564,1574,1581,1591,1597,1609,1614,1626,1631,1639,1647,1658,1664,1664,1675,1681,1695,1697,1709,1714,1714,1727,1731,1731,1743,1747,1748,1758,1764],[1951,1966,1969,1989,1997,2008,2014,2025,2031,2031,2041,2047,2058,2065],[2391,2407,2410,2415,2425,2431,2431,2441,2448,2458,2464,2475,2481,2491,2499,2508,2514,2525,2531,2543,2548,2561,2564,2575,2581,2590,2598,2608,2615,2628,2632,2644,2648,2648,2656],[2849,2864,2881,2882,2892,2898,2908,2914,2925,2931,2939,2948,2948,2959,2965,2975,2981,2992,3008],[3124,3130,3133,3142,3148,3159,3165,3175,3181,3182,3192,3198,3198,3209,3214,3225,3231,3242,3248,3259,3266,3277,3281,3292,3299,3309,3315,3326,3331,3344,3348,3359,3365,3365,3376,3381,3382,3395,3398,3409,3415,3425,3431,3442,3448,3457,3465,3465,3473,3481,3492,3498,3510,3515],[3701,3715,3719,3720,3725,3732,3742,3748,3759,3765,3765,3776,3781,3792,3798,3811,3815,3826,3831,3842,3848,3859,3865,3876,3883,3890],[4238,4265,4274,4281,4292,4300,4309,4315,4326,4332,4343,4348,4357,4365,4375],[4538,4565,4575,4582,4592,4599,4609,4615,4626,4633,4643,4649,4662,4665,4676],[4913,4932,4941,4948,4959,4965,4976,4982,4990,4999,5009,5015,5024,5032,5042,5049,5061,5065,5076],[5293,5308,5311,5312,5317,5326,5332,5343,5350,5360,5366,5376,5382,5393,5399,5399,5410,5415,5427,5432,5432,5443,5449,5449,5460,5467,5476,5482],[6246,6253,6257,6261,6266,6277,6282,6283,6294,6299,6310,6316,6316,6327,6332,6343,6349,6360,6366,6366,6377,6383,6394,6399,6411,6416,6424],[11642,11651,11662,11668,11679,11684,11696,11701,11701,11713,11718,11718,11729,11734,11735,11744,11751,11762,11768,11779,11785,11796,11802,11812,11820,11821,11831,11835,11846,11851,11860,11868,11879,11884,11896,11901,11902,11913,11918],[12136,12144,12151,12178,12185,12195,12203,12212,12218,12229,12235,12246],[12682,12697,12701,12710,12718,12729,12735,12746,12751,12760,12768,12779,12785,12796,12801,12802,12813,12821,12832,12835,12846],[13154,13165,13168,13169,13177,13185,13196,13202,13202,13214,13218,13230,13235,13235,13247,13252,13263,13268,13280],[13471,13484,13489,13490,13496,13502,13513,13518,13530,13535,13536,13543,13552,13562,13570,13580,13585,13588,13597,13602,13613,13619,13630,13635,13647,13652,13652,13664,13669,13669,13681,13685,13686,13697,13703,13713,13719,13730,13735,13747,13752,13763,13769,13780,13785,13786,13797,13802,13803,13814,13821,13822,13832,13835,13836,13847],[14154,14163,14166,14169,14181,14186,14197,14202,14214,14219,14230,14235,14236,14248,14252,14263,14269,14269,14281,14286,14286,14297,14302,14313,14319],[15192,15211,15219,15231,15236,15247,15252,15253,15265,15269,15281,15286,15286,15298,15302,15311,15319,15330,15336,15344,15362,15369],[15583,15593,15597,15598,15604,15614,15619,15631,15636,15636,15648,15653,15664,15669,15681,15686,15697,15703,15714,15719,15727,15736,15748],[16060,16074,16077,16083,16087,16094,16095,16104,16111,16120,16130,16136,16148,16153,16154,16165,16169,16170,16182,16186,16198,16203,16203,16215,16219,16220,16232,16236,16237,16248],[16530,16546,16549,16555,16564,16570,16581,16587,16598,16603,16603,16613,16620,16632],[16801,16816,16819,16821,16834,16836,16848,16854,16865,16870,16881,16886,16895,16903,16914,16920,16931,16936,16948,16953,16954,16963,16970,16970,16980,16988,16998,17004,17015,17020,17032,17036,17048,17053,17065,17070,17081,17086,17087,17097,17103,17115,17120,17121,17130,17136,17137,17147,17153,17165,17170,17182,17187,17198],[17416,17424,17427,17437,17448,17454,17465,17470,17482,17487,17498,17504,17515,17520,17532,17537,17537,17547,17553,17554,17564,17570,17570,17580,17587]],"version":"2.0.0"} y esta ley puede extenderse a cualquier número de sucesos de forma similar.
    • Se dice que dos sucesos son independientes entre sí si la ocurrencia de uno no afecta a la ocurrencia de otro, y lo mismo ocurre con cualquier número de sucesos.
    • Si dos sucesos son independientes entre sí, su regla de Producto viene dada por PAB=PAPB{"x":[[148,148,148,148,148,149,149,152,154,156,157,158,159,159,160,159,158,158,157,156,155],[166,164,162,159,157,156,153,151,150,150,151,155,158,168,171,175,183,188,192,195,203,210,214,217,218,217,213,208,205,196,192,186,184,182],[271,270,267,265,261,258,255,248,236,228,223,221,217,217,219,221,226,228,235,237,238],[263,260,259,258,258,260,261,263,266,268,271,276,278,281,287,290,295,300,302,307,311,314,316,317,318,319,319,318,317,315,314,313,313,312,312,312,311],[267,266,267,269,271,281,289,293,303,305,310,311,313],[365,363,363,363,364,366,369,370,372,374,379,383,389,391,397,402,407,410,416,418,423,425,427,428,429,429,429,429,428,428,428,428],[468,467,466,466,466,467,468,469,469,470,468,468,467,466,466],[462,460,460,459,463,465,472,477,484,487,490,496,501,503,506,507,508,509,509,509,507,503,499,495,494,489,488,485,485,486,487,488,491,493,495,497,501,506,509,512,513,513,512,508,506,498,494,484,480,472],[533,537,544,546,552,554,557,561,563,565,566,567,566,564,563,557,555,548,545,537,534,530],[612,613,616,621,627,632,636,638,643,644,646],[589,588,588,588,590,594,600,607,611,614,620,623,628],[677,678,678,679,680,680,680,680,679,678,677,676,676,675,674,673,673,673,673,674],[664,664,666,667,670,675,682,684,687,689,690,693,694,695,695,696,695,694,691,689,681,679,676,672,671],[751,749,747,746,744,743,742,738,736,731,726,721,719,714,713,711,711,712,713,714],[769,768,768,768,771,774,776,783,788,792,797,801,802,803,808,809,810,810,810,810,810,810,811,811,811,811,811,811,810,810,810],[775,775,775,775,775,775,775,776,780,784,787,794,801,810,814,818,819],[850,853,857,859,861,865,866,870,872,873,873,874,874,873,871,869,867,860,855,852,848,844],[904,904,903,903,903,904,904,904,903,903,902,901],[905,905,904,904,907,909,916,918,921,927,930,933,936,941,943,944,947,948,948,947,945,943,941,935,933,924],[1000,999,997,996,993,991,989,985,982,978,974,971,970,969,968,968,969,971,972,976,979,981,987,990,993,997,1000],[1029,1029,1029,1029,1029,1029,1029,1028,1028,1026,1025,1023,1023,1022,1021,1020,1020,1019],[1019,1019,1019,1020,1023,1025,1030,1032,1038,1043,1047,1050,1052,1052,1051,1050,1046,1044,1039,1037,1035,1034,1032,1031,1031,1032,1033,1035,1037,1040,1042,1045,1046,1047,1048,1047,1047,1044,1043,1041,1038,1036,1034,1029,1027,1025,1022,1019],[1089,1093,1094,1097,1101,1106,1109,1110,1112,1112,1111,1110,1104,1097,1090,1080,1075,1065,1061]],"y":[[239,238,237,248,254,261,270,299,321,343,368,382,416,427,453,459,472,475,476,475,471],[250,249,248,245,244,243,240,236,234,231,230,225,223,217,215,214,211,211,211,212,215,220,226,234,243,256,266,277,282,297,302,311,313,315],[220,215,214,215,220,223,228,242,270,292,311,321,352,361,388,396,414,418,425,426,426],[374,367,364,351,347,330,323,316,302,294,287,273,268,263,256,253,251,252,254,259,267,277,289,301,313,326,331,349,355,369,373,377,383,384,386,387,387],[323,322,319,318,317,313,310,309,306,305,303,303,303],[366,363,361,357,351,341,329,322,315,307,291,278,266,261,254,249,246,245,248,250,262,268,286,299,311,323,334,343,352,359,364,368],[261,263,265,266,269,278,283,302,308,321,345,352,363,368,372],[267,261,258,253,246,244,239,237,235,235,234,235,236,238,241,244,246,252,255,258,266,278,286,294,297,306,310,316,318,324,325,326,328,329,330,331,332,334,336,341,344,348,353,359,362,370,373,380,381,384],[214,209,210,213,219,223,228,239,245,259,275,289,297,318,324,343,349,366,371,385,388,394],[287,286,283,283,283,283,283,283,284,284,285],[334,333,332,331,330,329,327,325,324,324,322,322,322],[243,241,240,242,248,253,258,273,290,307,323,337,343,355,365,373,376,382,384,385],[277,269,260,257,255,250,246,245,245,246,248,252,254,256,261,264,269,274,279,282,291,294,297,301,303],[253,253,252,251,251,251,251,254,257,265,277,292,300,325,333,356,363,378,381,383],[343,333,320,316,302,289,284,268,258,252,247,244,244,244,255,260,272,279,287,300,306,318,328,336,339,347,349,352,357,358,359],[307,306,305,304,303,302,301,301,299,298,297,296,295,295,295,295,295],[228,223,222,223,225,230,234,242,253,260,267,274,295,301,313,319,324,340,351,359,365,370],[248,254,267,286,292,310,316,331,341,351,359,365],[267,262,255,253,249,247,244,243,242,242,242,243,244,247,249,250,254,256,261,266,273,276,280,287,290,299],[236,233,233,234,238,241,245,255,262,276,292,308,315,322,335,341,355,358,361,366,367,368,369,370,370,369,368],[267,265,266,268,276,280,286,299,306,326,332,345,349,352,358,361,363,365],[279,274,271,268,265,264,261,260,258,258,258,260,262,264,271,274,282,285,294,297,299,302,306,308,309,313,315,317,318,321,323,326,328,334,337,345,347,353,355,357,359,361,362,364,365,365,366,366],[234,232,232,236,244,255,268,275,299,307,332,340,362,376,390,403,410,421,426]],"t":[[0,5,10,37,42,43,54,59,71,76,87,93,104,109,120,126,137,144,154,159,171],[940,945,956,960,962,970,976,987,994,1003,1010,1020,1026,1037,1043,1043,1053,1059,1060,1070,1076,1087,1093,1103,1110,1120,1127,1136,1143,1153,1160,1170,1176,1180],[1361,1378,1381,1386,1393,1393,1404,1410,1420,1427,1437,1443,1453,1460,1468,1476,1485,1493,1503,1510,1514],[1737,1746,1751,1753,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    • Elcomplemento de un suceso, digamos A, es el subconjunto del espacio muestral que no está contenido en el propio A. Se denomina Ac. Se denota porAc. Un suceso y su complemento están relacionados por la ecuación , PA=1-PAC.
    La Regla del Producto de las Probabilidades La Regla del Producto de las Probabilidades
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    Preguntas frecuentes sobre La Regla del Producto de las Probabilidades
    ¿Qué es la Regla del Producto?
    La Regla del Producto en probabilidades establece que la probabilidad de que ocurran dos eventos independientes es el producto de sus probabilidades individuales.
    ¿Cómo se aplica la Regla del Producto?
    Para aplicar la Regla del Producto, multiplica las probabilidades de los eventos individuales: P(A y B) = P(A) * P(B).
    ¿Cuándo se usa la Regla del Producto?
    Se usa cuando se quiere encontrar la probabilidad de que ocurran dos o más eventos independientes de forma simultánea.
    ¿La Regla del Producto solo se aplica a eventos independientes?
    Sí, solo se aplica a eventos independientes, aquellos cuya ocurrencia de uno no afecta la probabilidad del otro.
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