680 CHAPTER 11 Probability Expected Value Expected value, also called expectation, is often used to determine the expected results of an experiment or business venture over the long term. People use expectation to make important decisions in many different areas. For example, expectation is used in business to predict future profits of a new product. The insurance industry uses expectation to determine premiums so insurance companies make a profit. Expectation is also used to predict the expected gain or loss in games of chance. Consider the following situation. Tim tells Barbara that he will give her $1 if she can roll an even number on a single die. If she fails to roll an even number, she must give Tim $1. Who would win money in the long run if this game were played many times? In this situation, we would expect in the long run that half the time Tim would win $1 and half the time he would lose $1; therefore, Tim would break even. Mathematically, we could determine Tim’s expected gain or loss by the following procedure: ( ) ( ) = ⎛ ⎝⎜ ⎞ ⎠⎟ ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ = + − = P P Tim’s expected gain or loss Tim wins amount Timwins Tim loses amount Timloses 1 2 $1 1 2 $1 $0 Note that the loss is written as a negative number. This procedure indicates that Tim has an expected gain or loss (or expected value) of $0. The expected value of zero indicates that he would indeed break even, as we had anticipated. A game (such as this one) with an expected value of 0 is called a fair game. If Tim’s expected value were positive, it would indicate a gain; if negative, a loss. The expected value, E, is calculated by multiplying the probability of an event occurring by the net amount gained or lost if the event occurs. If there are a number of different events and amounts to be considered, we use the following formula. Expected Value = ⋅ + ⋅ + ⋅ +…+ ⋅ E P P P P A A A A n n 1 1 2 2 3 3 The symbol P1 represents the probability that the first event will occur, and A1 represents the net amount won or lost if the first event occurs. P2 is the probability of the second event, and A2 is the net amount won or lost if the second event occurs, and so on. The sum of these products of the probabilities and their respective amounts is the expected value. The expected value is the average (or mean) result that would be obtained if the experiment were performed a great many times. Example 1 Expected Attendance For an outdoor concert, event organizers estimate that 20,000 people will attend if it is not raining and 12,000 people will attend if it is raining. On the day of the concert, meteorologists predict a 30% chance of rain. Determine the expected number of people who will attend this concert. Solution The amounts in this example are the number of people who will attend the concert. Since there is a 30% chance of rain on the day of the concert, the chance it will not rain is − = 100% 30% 70%. When written as probabilities, 30% and 70% are 0.3 and 0.7, respectively. PeopleImages.com - Yuri A/Shutterstock
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