document.write( "Question 1157959: I am learning about the probability of independent events. I just need to know why do we multiply the probabilities. I want to be able to explain the reason if I am asked. \n" ); document.write( "
Algebra.Com's Answer #780943 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are exceptions to the rules \r\n" );
document.write( "\"AND means multiply probabilities\" \r\n" );
document.write( "and \r\n" );
document.write( "\"OR means add probabilities\".  \r\n" );
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document.write( "What you asked for is WHY this always works:\r\n" );
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document.write( "P(A and B) = P(A)∙P(B [when we know that A is the case]) =\r\n" );
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document.write( "P(A and B) = P(B)∙P(A [when we know that B is the case]) \r\n" );
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document.write( "If B is the case a third of time, and A is the case half of the time (that is,\r\n" );
document.write( "when we know that B is the case), then half of those times when B is the case, A\r\n" );
document.write( "will also be the case. So they both are the case simultaneously half of a third\r\n" );
document.write( "of the time, so we multiply their probabilities and get (1/3)(1/2) = 1/6, so\r\n" );
document.write( "they both will be the case AT THE SAME TIME 1/6th of the time. \r\n" );
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document.write( "Caution: When A rules B out, we may NOT multiply their individual probabilities,\r\n" );
document.write( "for the probability that they are BOTH the case is ZERO, because they cannot\r\n" );
document.write( "both happen at the same time. \r\n" );
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document.write( "Now we need to know the three kinds of pairs of events that are talked about in\r\n" );
document.write( "probability studies.  We must learn and understand and be able to distinguish\r\n" );
document.write( "them:\r\n" );
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document.write( "1. A pair of (mutually) INDEPENDENT events\r\n" );
document.write( "2. A pair of (mutually) DEPENDENT events\r\n" );
document.write( "3. A pair of MUTUALLY EXCLUSIVE events.\r\n" );
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document.write( "[The word \"mutually\" is seldom used in the first two cases but is always used\r\n" );
document.write( "in the third case.]\r\n" );
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document.write( "If a pair of events are such that one event DOES NOT INCREASE or DECREASE the\r\n" );
document.write( "probability of the other event, then the events are INDEPENDENT.\r\n" );
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document.write( "If a pair of events are such that one event INCREASES or DECREASES the\r\n" );
document.write( "probability of the other event, then the events are DEPENDENT.\r\n" );
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document.write( "A special case of a pair of DEPENDENT events is the case when one event\r\n" );
document.write( "DECREASES the probability of the other event all the way to ZERO. In other words\r\n" );
document.write( "one event COMPLETELY RULES THE OTHER EVENT OUT!  Then the pair of events are\r\n" );
document.write( "MUTUALLY EXCLUSIVE. \r\n" );
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document.write( "[For this last case, think of the less common use of \"EXCLUSIVE\" as 'EXCLUDING'.\r\n" );
document.write( "One event EXCLUDES the other] \r\n" );
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document.write( "The two formulas that ALWAYS work in ALL three cases are\r\n" );
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document.write( "(1)   P(A or B) = P(A) + P(B) - P(A and B)\r\n" );
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document.write( "(2)   P(A and B) = P(A)∙P(B|A) = P(B)∙P(A|B), where the \"|\" means \"given\".\r\n" );
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document.write( "In the case of a pair of INDEPENDENT events, formula (2) above can be\r\n" );
document.write( "simplified.\r\n" );
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document.write( "In the case of a pair of MUTUALLY EXCLUSIVE events, formula (1) above can be\r\n" );
document.write( "simplified: \r\n" );
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document.write( "In cases of pairs of INDEPENDENT events A, B,\r\n" );
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document.write( "P(A given B) = P(A|B) = P(A)\r\n" );
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document.write( "and \r\n" );
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document.write( "P(B given A) = P(B|A) = P(B)\r\n" );
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document.write( "so (2) above simplifies to \r\n" );
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document.write( "(2a)   P(A and B) = P(A)∙P(B) in that case.\r\n" );
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document.write( "In the case of pairs of MUTUALLY EXCLUSIVE events,\r\n" );
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document.write( "P(A and B) = 0, so (1) simplifies in that case to\r\n" );
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document.write( "(1a)   P(A or B) = P(A) + P(B)\r\n" );
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document.write( "Caution: Don't get any of the three cases mixed up!  And, I repeat, remember\r\n" );
document.write( "that there are exceptions to the rules \"AND means multiply probabilities\" and\r\n" );
document.write( "\"OR means add probabilities\".  \r\n" );
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document.write( "Edwin
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