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Unformatted text preview: MA/STAT416: Probability Lecture Notes Spring 2011 1 Chapter 3 January 26, 2011 3 Conditional Probability and Independence 3.1 Basic Formulas Basic Examples Example 1 . Find the conditional probability that the sum of two rolls of a fair die is 12, if the first roll is known to be six. Example 2 . Draw a card from a deck of 52 cards, what is the probability of getting a heart? Suppose now we already know that the card is not a club, what is the probability of getting a heart? If we dont know any other information, the probability is 13 52 . Now, given that the card is not a club, the probability becomes 1 3 . Conditional probability is used to model the probability of events when certain in- formation is provided. Definition 3. For any two events A and B with P ( B ) > 0, the conditional proba- bility of A given B has occurred denoted P ( A | B ), is defined by P ( A | B ) = P ( A B ) P ( B ) . General Multiplication Rule Rearranging the terms, we get the product rule for P ( A B ) : P ( A B ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) . Example 4 . Let A , B be two events with P ( A ) = 0 . 5 and P ( B | A ) = 0 . 8. What is the probability that A and B will both happen? Example 5 . Let A be the event of getting a heart, B be the event of getting a 4. What is the conditional probability of A given B ? P ( A | B ) = P ( A B ) P ( B ) = 1 / 52 4 / 52 = 1 4 . Example 6 . Deal 2 cards from a deck of 52 cards (well shuffled, without replacement)....
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