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Expected Value

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1) In a particular lottery, 3 million tickets are sold each week for 50c apiece. Out of the 3 million tickets, 12,006 are drawn at random and without replacement and awarded prizes: twelve thousand $25 prizes, four $10,000 prizes, one $50,000 prize, and one $200,000 prize. If you purchases a single ticket each week, what is the expected value of this game to you?

2) Let the random variable X be the number of days that a certain patient needs to be in the hospital. Suppose X has the p.m.f. f(x) =(5-x)/10, x = 1,2,3,4. If the patient is to receive $200 from an insurance company for each of the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization?

3) In the gambling game chuck-a-luck, for a $1 bet it is possible to win $1, $2, or $3 with respective probabilities 75/216, 15/216, and 1/216. Let X equal the payoff for this game and find E(X). Note that when a bet is won, the $1 that was bet, in addition to the $1, $2, or $3 that is won, is returned to the bettor.

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The solution contains the determination of the expected value of a random variable in a number of different situations.

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