Lecture 6 notes - 4.3 CONDITIONAL PROBABILITY Ref Devore...

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4.3 CONDITIONAL PROBABILITY Ref: Devore 7th Ed. Sections 2.4, 2.5. Suppose we toss a fair coin 5 times and it comes up heads all 5 times. Is the next toss more likely to be a tail? Ans: No. P [6 heads in a row | 5 heads in a row ] = 1 2 Definition . The conditional prob of event A given that event B has occurred is P ( A | B ) = P ( A B ) /P ( B ) Multiplication rule (Devore 7e, p. 69) P ( A B ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) . Often P ( A | B ) or P ( B | A ) is known, and this rule allows us to calculate P ( A B ) . Example: Draw 2 cards from a pack. What is the probability both are hearts? A = 1st card is heart , B = 2nd card is heart . P (2 hearts) = P ( A B ) = P ( A ) P ( B | A ) = 13 52 × 12 51 = ˆ 13 2 ! / ˆ 52 2 ! Can extend multiplication rule. E.g. P ( A B C ) = P ( A ) P ( B | A ) P ( C | A B ) . e.g. Prob(5 hearts) = 13 52 × 12 51 × 11 50 × 10 49 × 9 48 = ˆ 13 5 ! / ˆ 52 5 ! Independence of Events We say events A and B are independent if P ( A B ) = P ( A ) P ( B ) . Rationale: If A,B independent, this implies P ( A | B ) = P ( A ) and P ( B | A ) = P ( B ) .
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