684 CHAPTER 11 Probability This answer, −$3.00, checks with the answer obtained by multiplying the expectation for a single ticket by 2. Let’s look at one more example in which a person must pay for a chance to win a prize. In the following example, there will be more than two amounts to consider. Example 7 Raffle Tickets One thousand raffle tickets are sold for $1 each. One grand prize of $500 and two consolation prizes of $100 will be awarded. The tickets are placed in a bin. The winning tickets will be selected from the bin. Assuming that the probability that any given ticket selected for the grand prize is 1 1000 and the probability that any given ticket selected for a consolation prize is , 2 1000 determine a) Irene’s expectation if she purchases one ticket. b) Irene’s expectation if she purchases five tickets. Solution a) Three amounts are to be considered: the net gain in winning the grand prize, the net gain in winning one of the consolation prizes, and the loss of the cost of the ticket. If Irene wins the grand prize, her net gain is $499 ($500 minus $1 spent for the ticket). If Irene wins one of the consolation prizes, her net gain is $99 ($100 minus $1). The probability that Irene wins the grand prize is 1 1000 and the probability that she wins a consolation prize is . 2 1000 The probability that she does not win a prize is − − = 1 . 1 1000 2 1000 997 1000 = ⋅ + ⋅ + ⋅ = + + − = + − = − = − E P A P A P A 1 1000 (499) 2 1000 (99) 997 1000 ( 1) 499 1000 198 1000 997 1000 300 1000 0.30 1 1 2 2 3 3 Thus, Irene’s expectation is −$0.30 per ticket purchased. b) On average, Irene loses 30 cents on each ticket purchased. On five tickets, her expectation is −( 0.30)(5), or −$1.50. ■ Now try Exercises 37 Timely Tip In any expectation problem, the sum of the probabilities of all the events should always be 1. Note in Example 7 that the sum of the probabilities is 1 1000 2 1000 997 1000 1000 1000 1. + + = = Fair Price In Example 6, we determined that Josh’s expectation was −$1.50. Now we will determine the price that should have been charged for a ticket so that his expectation would be $0. If Josh’s expectation were to be $0, he could be expected to break even over the long run. Suppose that Josh paid 50 cents, or $0.50, for the ticket. His expectation, if paying $0.50 for the ticket, would be calculated as follows. = ⋅ + ⋅ = + − = − = P P Expectation (Josh wins) (amount won) (Josh loses) (amount lost) 1 100 (49.50) 99 100 ( 0.50) 49.50 100 49.50 100 0 Thus, if Josh paid 50 cents per ticket, his expectation would be $0. The 50 cents, in this case, is called the fair price of the ticket. The fair price is the amount to be paid that will result in an expected value of $0. The fair price may be determined by adding the cost to play to the expected value . Applets Simulation Raffle Winnings StatCrunch
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