stat249-lec3-1.pdf - STAT 249 Lecture 3 part 1 1...

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STAT 249 - Lecture 3 part 1 1 Calculating Probabilities using Conditional Probabilities - (Sec 2.8- 2.9) We discussed how to calculate conditional probabilities if we use the probabilities of the intersection of two events in question. However, in practice we can often more easily calculate in reverse: use conditional probabilities to calculate the probabilities of intersection of two events. The defining formulae for P ( A | B ) and P ( B | A ) imply that P ( A B ) is: P ( A B ) = P ( A | B ) · P ( B ) = P ( B | A ) · P ( A ) Example 1 Suppose we roll two dice, A =first die is an even #, B =sum of the two dice is even #; P ( A ) = 1 / 2 is easy to calculate, and given that the first die is even, the sum will be even if the second die is even as well which has a chance 1 / 2 , so the conditional probability P ( B | A ) = 1 / 2 . So the chance that first die is even and the sum of two dice is even is P ( A B ) = (1 / 2)(1 / 2) = 1 / 4 . Of course this also follows if we use C =second die is even and re-express the intersection as A B = A C so P ( A B ) = P ( A C ) = P ( A ) P ( C ) because the outcomes on the two dice are independent implies A and C are independent events.

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