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471takehome - that at the time one becomes empty the other...

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Prelim I, 6 hours SHOW ALL WORK! (1) [10 pts] Events A,B,C are independent. (a) Show that A c ,B c ,C c are independent. (b) Show P ( A B C ) = 1 - (1 - P ( A ))(1 - P ( B ))(1 - P ( C )) . (2) [10 pts] You have 9 normal coins and one trick coin with two heads. You pick one at random and flip it 3 times getting heads each time. What is the probability that you picked the trick coin? (3) [15 pts] There are 1000 raffle tickets. 100 of them are winning tickets and lead to prizes. Suppose you buy 3 raffle tickets. If X is a random variable representing the number of prizes you win, what is E ( X )? In other words, what is the expected number of prizes you should get? (4) [25 pts] You pick six cards out of a deck of 52. (a) What is the probability that exactly 5 of them are the same suit? (b) What is the probability that you get at least one card of each suit? (5) [15 pts] There are three urns with 10 balls each. Suppose we choose an urn at random and draw a ball from that urn, We continue this process until one urn is empty. What is the probability
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Unformatted text preview: that at the time one becomes empty, the other two each have 3 balls left? What is the probability that at the time one becomes empty, one urn has 2 balls left and the other 3 balls left? (6) [25 pts] Suppose the number of customers arriving in a store during a typical hour is Poisson distributed with parameter λ . Let p be the typical fraction of customers that are female. In other words, conditioning on the event that exactly n customers come during a given hour, the probability that k of them are female is n ! k !( n-k )! p k (1-p ) n-k . (a) Show that the number of female customers arriving in a given hour is Poisson distributed with parameter pλ . (b) Show that the event that no females arrive during the hour is independent of the event that no male arrives during the hour....
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