Lecture-5

Lecture-5 - Lecture 5. Conditional Probability and...

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Unformatted text preview: Lecture 5. Conditional Probability and Independence Definition. Suppose ( , P ) is a probability space and A is an event such that P ( A ) negationslash = 0. Let B be another event. We define the conditional probability of B given A denoted P ( B | A )by: P ( B | A ) := P ( A B ) P ( A ) . Example. Ted and Alice have two children. Given that they have a boy, what is the probability that they have two boys? Solution. With no knowledge of the genders of their children, we would assume that gg , gb , bg and bb are equally probable, since there are equal chances of having a girl or a boy as first child, and the same is true for the second regardless of the gender of the first. But since we know they have a boy, only gb , bg and bb are possible. These being equally probable, the answer must be 1 / 3. ///// Comment. To connect the example to the definition, we name the events that come into play. Call the event of having a boy B . Then B = { gb, bg, bb } . Call the event of having two boys A . Then A = { bb } , and A B = { bb } . P ( B | A ) = P ( A B ) P ( A ) = 1 / 4 3 / 4 = 1 3 . Comment. This problem seems to violate the requirement that an experiment is a re- peatable action. One might argue that Ted and Alice either have two boys or they dont, so the probability they have two boys is either 1 or 0, and we just dont know which. In questions of this type, the reader is implicitly asked to view Ted and Alice as a random cou- ple. A different perspective is sometimes taken, namely the subjectivist interpretation of probability. In this view, probability statements are statements of quantified certaintyprobability....
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Lecture-5 - Lecture 5. Conditional Probability and...

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