11.3 Expected Value (Expectation) 685 In Example 6, the cost to play was $2 and the expected value was determined to be −$1.50. The fair price for a ticket in Example 6 may be determined as follows. = + − + = Fair Price expected value cost to play = 1.50 2.00 0.50 We obtained a fair price of $0.50. If the tickets were sold for the fair price of $0.50 each, Josh’s expectation would be $0, as shown above. In our next example, we will determine the fair price of the raffle tickets sold in Example 7. Did You Know? Expectation and Playing the Lottery How realistic is it that you will win a lottery jackpot? Expectation can be used to provide some insight into this question. We will use the Florida Lottery game Lotto to demonstrate the expectation of a lottery ticket. A Lotto ticket costs $2, and players select 6 different numbers from 1 through 53. To win the jackpot, a player must match all 6 numbers that are randomly drawn from a bin. Although the prize amounts vary, the minimum jackpot is worth $1,000,000. A player who matches 5 of the 6 numbers wins $3000, 4 of the 6 numbers wins $50, and 3 of the 6 numbers wins $5. In Section 11.8 and 11.9, we will show how the probabilities of winning each prize can be calculated. The following calculation shows the expectation for a Lotto ticket using these minimum prize amounts minus the $2 ticket price. = + + + + = + + + + − = ≈ − PA P A P A P A P A Expected value 1 22,957, 480 (999,998) 282 22,957, 480 (2998) 16,215 22,957, 480 (48) 324,300 22,957, 480 (3) 22,616,682 22,957, 480 ( 2) 41,636,710 22,957, 480 1.81 1 1 2 2 3 3 4 4 5 5 Thus, the expectation of purchasing a Lotto ticket is −$1.81. This means in the long run, players lose $1.81 cents for each Lotto ticket they buy. Moreover, the calculation does not consider having to pay taxes on the winnings or having to split the jackpot if there are multiple winners. Someone might argue that “Somebody will win!” While this may be true, a negative expectation indicates that purchasing lottery tickets is not a wise investment! Fair Price To determine the fair price, use the following formula. = + Fair price expected value cost to play Example 8 Fair Price of a Raffle Ticket In Example 7, the cost of a raffle ticket was $1, and we determined that the expected value was −$0.30. a) Determine the fair price for this raffle ticket. b) Verify that if the raffle tickets were sold for the fair price determined in part (a), the expectation would be $0.00. Solution a) We use the formula = + = − + = Fair price expected value cost to play 0.30 1.00 0.70 Thus, the fair price for the raffle ticket is $0.70, or 70 cents. b) When calculating the expected value, the probabilities will be the same as in Example 7, only the net amounts change. If the raffle tickets were sold for $0.70, then A ,1 the net amount gained by winning the grand prize would be − $500 $0.70, or $499.30. A ,2 the net amount gained by winning one of the consolation prizes would be − $100 $0.70, or $99.30. A ,3 the amount lost by not winning a prize is the price of the ticket, or −$0.70. Thus, we have P A P A P A Expected value 1 1000 (499.30) 2 1000 (99.30) 997 1000 ( 0.70) 499.3 1000 198.6 1000 697.9 1000 499.3 198.6 697.9 1000 0 1000 0 1 1 2 2 3 3 = + + = + + − = + − = + − = = Thus, if the raffle tickets were sold for $0.70, the expectation would be $0.00. ■ Now try Exercise 45 Example 9 Expectation and Fair Price At a game of chance, the expected value is determined to be −$1.50, and the cost to play the game is $4.00. Determine the fair price to play the game.
RkJQdWJsaXNoZXIy NjM5ODQ=