11.3 Expected Value (Expectation) 683 Now we will consider a problem similar to Example 5, but this time we will assume that Josh must purchase the ticket for the door prize. Thus, Jayden has an expectation of $8750 for this listing. This means if Jayden has more listings like this one, with these probabilities and amounts, in the long run Jayden would average a net gain of $8750 per house. However, it is important to remember that there is a 25% chance that Jayden will lose $2500 on this listing or on similar listings with these probabilities and amounts. ■ Now try Exercise 39 Example 5 Winning a Door Prize When Josh attends a charity event, he is given a free ticket for the $50 door prize. A total of 100 tickets will be given out. Determine his expectation of winning the door prize. Solution The probability of winning the door prize is , 1 100 since Josh has 1 of 100 tickets. If he wins, his net or actual winnings will be $50, since he did not pay for the ticket. The probability that Josh loses is . 99 100 If Josh loses, the amount he loses is $0 because he did not pay for the ticket. = ⋅ + ⋅ = + = = P P Expectation (Josh wins) (amount won) (Josh loses) (amount lost) 1 100 (50) 99 100 (0) 50 100 0.05 Thus, Josh’s expectation is $0.50, or 50 cents. ■ Now try Exercise 33 Example 6 Winning a Door Prize When Josh attends a charity event, he is given the opportunity to purchase a ticket for the $50 door prize. The cost of the ticket is $2, and 100 tickets will be sold. Determine Josh’s expectation if he purchases one ticket. Solution As in Example 5, Josh’s probability of winning is . 1 100 However, if he does win, his actual or net winnings will be $48. The $48 is obtained by subtracting the cost of the ticket, $2, from the amount of the door prize, $50. There is also a probability of 99 100 that Josh will not win the door prize. If he does not win the door prize, he has lost the $2 that he paid for the ticket. Therefore, we must consider two amounts when we determine Josh’s expectation, winning $48 and losing $2. = ⋅ + ⋅ = + − = − = − = − P P Expectation (Josh wins) (amount won) (Josh loses) (amount lost) 1 100 (48) 99 100 ( 2) 48 100 198 100 150 100 1.50 Josh’s expectation is −$ 1.50 when he purchases one ticket. ■ Now try Exercise 35 In Example 6, we determined that Josh’s expectation was −$1.50 when he purchased one ticket. If he purchased two tickets, his expectation would be − 2( $1.50), or −$3.00. We could also compute Josh’s expectation if he purchased two tickets as follows: = + − = − E 2 100 (46) 98 100 ( 4) 3.00
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