196 CHAPTER 4 Discrete Probability Distributions Expected Value The mean of a random variable represents what you would expect to happen over thousands of trials. It is also called the expected value. The expected value of a discrete random variable is equal to the mean of the random variable. Expected Value = E1x2 = m = ΣxP1x2 DEFINITION In most applications, an expected value of 0 has a practical interpretation. For instance, in games of chance, an expected value of 0 implies that a game is fair (an unlikely occurrence). In a profit-and-loss analysis, an expected value of 0 represents the break-even point. Although probabilities can never be negative, the expected value of a random variable can be negative, as shown in the next example. Finding an Expected Value At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. Find the expected value and interpret its meaning. SOLUTION To find the gain for each prize, subtract the price of the ticket from the prize. For instance, your gain for the $500 prize is $500 - $2 = $498 and your gain for the $250 prize is $250 - $2 = $248. Write a probability distribution for the possible gains (or outcomes). Note that a gain represented by a negative number is a loss. Gain, x $498 $248 $148 $73 -$2 Probability, P1x2 1 1500 1 1500 1 1500 1 1500 1496 1500 -$2 represents a loss of $2 Then, using the probability distribution, you can find the expected value. E1x2 = ΣxP1x2 = $498# 1 1500 + $248# 1 1500 + $148# 1 1500 + $73# 1 1500 + 1-$22 # 1496 1500 = -$1.35 Interpretation Because the expected value is negative, you can expect to lose an average of $1.35 for each ticket you buy. TRY IT YOURSELF 7 At a raffle, 2000 tickets are sold at $5 each for five prizes of $2000, $1000, $500, $250, and $100. You buy one ticket. Find the expected value and interpret its meaning. Answer: Page A38 EXAMPLE 7
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