682 CHAPTER 11 Probability P P (guesses correctly) 1 5 (guesses incorrectly) 4 5 = = P A P A Maria’s expectation 1 5 (2) 4 5 1 2 2 5 2 5 0 1 1 2 2 = ⋅ + ⋅ = + ⎛ − ⎝⎜ ⎞ ⎠⎟ = − = Thus, Maria’s expectation is zero when she guesses. b) Since Maria’s expectation when she guesses is zero, it is neither to her advantage nor to her disadvantage to guess. c) If Maria can eliminate one possible choice, one of four remaining choices will be the correct answer. P P (guesses correctly) 1 4 (guesses incorrectly) 3 4 = = = ⋅ + ⋅ = + ⎛ − ⎝ ⎞ ⎠ = − = − = P A P A Maria’s expectation 1 4 (2) 3 4 1 2 2 4 3 8 4 8 3 8 1 8 1 1 2 2 Thus, if Maria can eliminate one of the choices, her expectation is . 1 8 d) Since the expectation is a positive , 1 8 Maria will, on average, gain 1 8 point each time she guesses when she can eliminate one possible choice. ■ Guesses incorrectly 2 Guesses correctly 2 Guesses correctly 2 Guesses incorrectly 2 Now try Exercise 17 In the expected value formula, the amounts refer to net amounts , which are the actual amounts gained or lost. Our next three examples illustrate how net amounts are used in applications. Example 4 Real Estate Listing Jayden is a real estate broker who has a contract to sell a house. Such a contract is called a listing . Jayden estimates that this listing will cost her $2500 for various expenses including advertising. Jayden must pay these expenses whether she sells the house or not. If Jayden sells the house within 6 months, she will receive a commission of $15,000. If she does not sell the house within 6 months, the listing will expire, and she will not receive a commission, but she still must pay the $2500 for various expenses. Jayden estimates the probability that she sells the house within 6 months is 75%. Determine Jayden’s expectation for this listing. Solution There is a 75% probability that Jayden sells the house within 6 months; thus, = P 0.75. 1 If she sells the house within 6 months, she will gain the commission of $15,000 but will have $2500 in expenses. Thus, = − A $15,000 $2500, 1 or = A $12,500. 1 There is a 25% probability that Jayden does not sell the house within 6 months; thus, = P 0.25. 2 If she does not sell the house within 6 months, she will not receive a commission, but she still must pay the $2500 in expenses. Thus, = − A $2500. 2 Substituting P A P , , , 1 1 2 and A2 into the expected value formula gives us the following. = + = + − = PA P A Jayden’s expectation (0.75)($12,500) (0.25)( $2500) $8750 1 1 2 2 Did You Know? Expected Value and Decisions Expected value can be used to help evaluate the consequences of decisions. These decisions can range from routine decisions such as deciding where to park your car to life-changing decisions such as deciding whether to go back to college. For each decision we make, we must consider the probabilities and the gains or losses associated with each possible outcome. For example, if you decide to park your car illegally, the probability of getting caught may be low, but the loss associated with this decision, a parking ticket, may be high. If you decide to go back to college, there is a probability that you might not succeed. Associated with this probability is the loss of time and money for books and tuition. However, there is also a probability that you will succeed and graduate. Associated with this probability is the ability to attain a higherpaying and more rewarding career! Expected value can be used to help make wise decisions that help us in the long run. Michaeljung/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=