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elemprob-fall2010-page23

elemprob-fall2010-page23 - or P(F E = P(E)P(F which agrees...

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or P ( F E ) = P ( E ) P ( F ) , which agrees with the definition of independence we gave before. Let us give two more examples. An example: Suppose an urn holds 5 red balls and 7 green balls. You draw two balls without replacement. What is the probability the second ball is red? Answer. Let A be the event that the first ball is red, B that the second ball is, and we want P ( B ). Then P ( B ) = P ( A B ) + P ( A c B ) = P ( B | A ) P ( A ) + P ( B | A c ) P ( A c ) . The probability of A is 5 12 and the probability for A c is 7 12 . Given that the first ball is red, there are now 4 red balls and 7 green, so P ( B | A ) = 4 11 . Similarly, P ( B | A c ) = 5 11 . Therefore P ( B ) = 4 11 · 5 12 + 5 11 · 7 12 = 5 12 , which is what one would expect. An example. This is known as the Monty Hall problem after the host of the TV show of the 60’s called Let’s Make a Deal . There are three doors, behind one a nice car, behind each of the other two a bale of straw. You choose a door. Then Monty Hall opens one of the other doors, which shows a bale of straw. He gives you the opportunity of
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