L7_rev2 - Conditional Probability Probabilities do not live...

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Unformatted text preview: Conditional Probability Probabilities do not live in a vacuum. They are specified by conditions which may change or about which additional information may become available. Con- sider A,B with P ( A ) ,P ( B ) given. Sup- pose have knowledge that B occurred. How would the probability of A change? Notation. P ( A | B ) denotes the conditional probability of A given that B occurred. This is a new probability. How should it be defined? 1 Example. Shuffle cards and draw two cards at random. B = { first card is an Ace } , A = { second card is an Ace } . Natural to define P ( A | B ) = 3 51 What about P ( A B )? Using the Mul- tiplication Rule P ( A B ) = | A B | | B | = 4 3 52 51 = 4 52 3 51 = P ( B ) P ( A | B ) Makes sense. In order for A B to occur, B has to occur and then A has to occur. 2 Definition. The conditional probability P ( A | B ) is defined to be that number for which P ( A B ) = P ( B ) P ( A | B ) (1) This is equivalent to P ( A | B ) = P ( A B ) P ( B ) (2) which is how most texts define condi- tional probability. Observe that in ei- ther equation there are three quantities and any two give the third. Sometimes P ( A | B ) is provided from the context, as in (1), at other times it must be calculated, as in (2)....
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L7_rev2 - Conditional Probability Probabilities do not live...

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