03_01 - Conditional Probability Q: Should the following...

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p. 3-1 Conditional Probability Q : Should the following probabilities be different? Event = 王建民本季戰績至少獲得 6 次勝投 Event = rain tomorrow P =?? if no information about where you are staying P =?? if you are staying in a desert P =?? if a typhoon will hit the place you stay tomorrow Q : What causes the differences? For an event, new information (i.e., some other event has occurred) could change its probability We call the altered probability a conditional probability • Mathematical Definition: If A and B are two events in a sample space and P ( A )>0, then is called the conditional probability of B given A . P ( B | A ) P ( A B ) P ( A ) P =?? in the beginning of the season P =?? in the middle of the season P =?? now p. 3-2 In the classical model, P ( A ) = # A /# and P ( A Å B)=#( A Å B )/# Example: A family is known to have 2 children, at least one of whom is a girl . Q : Probability that the other is a boy=?? = { bb , bg , gb , gg } A ={ bg , gb , gg } and B ={ bb , bg , gb } P ( B | A ) = #( A Å B )/# A = 2/3 Note: # is reduced to # A . P ( B | A )= #( A B ) / # # A/ # = #( A B ) # A . In effect by conditioning, A, B B A. and, for an arbitrary event B in to occur when A has occurred, we need that both A and B occur together, i.e., we are restricting the sample space from to A , i.e., The division by P ( A ) in the definition above rescales
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03_01 - Conditional Probability Q: Should the following...

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