MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the population mean, variance, and standard deviation for the probability experiment: Toss a coin three times, and count the number of heads. The probabilities are ( Number of heads Probability 2 as we ve computed before. Theoretically, if we perform the experiment eight times, and the results came out exactly as the probabilities predict, we can compute the mean, variance, and standard deviation in the normal way (except we use the population formulas instead of the sample formulas. x x µ (x µ 2 (2.5 2.25.5.25.5.25.5.25 2.5.25 2.5.25 2.5.25.5 2.25 6. µ = =.5 σ2 = 6. =.75 σ =.75 =.7 Look at the mean first. We find that ( µ = + + + + 2 + 2 + 2 + = + + 2 + = + + 2 +
2 Note that the mean is simply the sum of each of the outcomes multiplied by their probabilities. If we theoretically performed the experiment times, we would get exactly the same thing. This is the official way that means are computed for probability distributions. In particular, (4 µ = ( x P(x Since the population variance is the mean of the deviations squared, we have a similar formula for it.. (5 σ 2 = ( (x µ 2 P(x and we still have σ = σ 2., 2. Expected Values We now have a formula for computing the mean of a probability distribution. (6 µ = ( x P(x. Another name for the mean of a probability distribution is the expected value. We use the symbols E(x for the expected value of a (random variable x. The formula for the expected value is, (7 E(x = ( x P(x of course. Remember that the expected value and mean are the same thing. Sometimes it makes no sense to talk about an expected value. For example, if we toss a coin, the possible outcomes are H or T. What is the expected value or average outcome? If we roll a die, the outcomes are, 2,, 4, 5, and 6. We can compute the expected value in this case. ( E(x = 6 + 2 6 + 6 + 4 6 + 5 6 + 6 6 = 2 6 =.5 This kind of makes sense, the formula actually works, but the number.5 doesn t tell us anything terribly meaningful. We can add meaning. Suppose this is a gambling game. We roll the die, and I ll give you $ if it comes up, $2 if it comes up 2, etc. Now, the expected value has more meaning. I ll give you $.5 on average. Statistically speaking, we, might say that being able to play the game is worth $.5. If I were a casino, I d probably charge you $4 to play. You d give me $4, and I d give you $.5 on average. That s a losing deal for you.
MA 5 Lecture 4 - Expected Values Gambling games give us a good way to think about expected values. Let s play another game. You bet $ and roll a pair of dice. If the dice come up, then you get your $ back, and you win $. Otherwise, you lose your dollar. We can describe this game with two outcomes. You win $ or you lose $. The probability of winning is the same as rolling a, that is P( = 6 =. If that s the probability of winning, then the probability of losing must be. The probability distribution comes out like this. (9 x P(x P( = P( = To compute the expected value of the winnings, we just follow formula (7. x P(x x P(x ( P( = = P( = = On average, this says, you lose $ or about eight cents. 2.. What about that $ bet? As you try to make sense of this stuff, the $ bet can be confusing. Let s say you come to the gaming table with $, you bet the dollar, and win. You ve got $ dollars now, right. But you re only ahead by $. The $ is your winnings. Can we do the computations with the $ in your hands? Yes. Let s say that it costs $ to play. If you win, the casino slides you $. If you lose, they simply ask you to play again. Now the outcomes are $ and $, with the same probabilities we saw before. Let s compute the expected value. x P(x x P(x ( Here s the interpretation. You paid your dollar. Now you re waiting for the dice to be rolled. How much money, on average, is the outcome of the game? Well, the expected outcome is $ or about $.92. You paid a dollar, but the game s only worth $.92. You re going to lose eight cents on average. This is the same conclusion as before.
4 2.2. What s the expected value of a lottery ticket? This second way of analyzing a gambling game fits well with evaluating lottery tickets. Let s say that you can buy a lottery ticket for $5. You bought it, and now your five dollars are gone. But, there are a thousand lottery tickets. One of them is worth $2,. Another is worth $,. Ten of the tickets are worth $. The rest are worth nothing. What is the expected value of your ticket? First we need to know the probabilities. Since there is one ticket worth $2,, we can say that the probability that this is your ticket is. The probability that the $, ticket is yours must also be. x P(x x P(x ( $2 $ $ $ 9 2 4 = $4 Each ticket s expected value is $4. Since you paid $5 for it, you lose a dollar on each ticket (on average. We could have computed the expected value on net winning. The outcomes would have been $995, $995, $95, and $5. You would get E(x = $.. Quiz 4. In Roulette, there are equally likely outcomes, {, 2,,..., 5, 6,, }. Let s say that you bet $ on one of the numbers. If your number comes up, you get your $ bet back plus $5. Otherwise, you lose your $ bet. Find the expected value of your winnings in dollars (with outcomes: win $5 or lose $. Don t forget the negative sign, if you need one. Round your answer to two decimal places. 2. In a lottery, there are 2 tickets. One is worth $5, two are worth $, twenty are worth $, and a hundred are worth $. The rest are worth nothing. Find the expected value of a ticket in dollars. Don t forget the negative sign, if you need one. Round your answer to two decimal places.
MA 5 Lecture 4 - Expected Values 5 4. Homework 4 Remember on a Roulette wheel, there are possible outcomes:, 2,..., 6,, (all equally likely. Half of the numbers -6 are colored red ( of them, and the rest are colored black. and are green.. What is the probability of a red number coming up? Express your answer as a fraction. 2. For a $ bet on red, if red comes up, you get your dollar back plus another $. Your winnings are either win $ or lose $. Find the expected value of your winnings in dollars. Don t forget the negative sign, if you need one. Round your answer to three decimal places.. It s also possible to bet $ on a group of four numbers. What is the probability of one of your numbers coming up? Express your answer as a fraction. 4. If one of your numbers comes up, you get your bet back plus $. Find the expected value of your winnings in dollars (your outcomes are +$ and $. Don t forget the negative sign, if you need one. Round your answer to three decimal places. 5. In a lottery, there are tickets. One is worth $, five are worth $, fifty are worth $, and five-hundred are worth $. The rest are worth nothing. What is the expected value of a ticket in dollars? Don t forget the negative sign, if you need one. Round your answer to two decimal places. 7 Quiz answers: 5 + ( = 2 =.526579 =.5 (you lose 5 cents on average. 2 5 2 + 2 2 + 2 2 + 2 = 7 2 =.65 (each ticket is worth $.65 on average. HW answers: = 9 2. 2 + ( 9 = 2 = $.5. 4 = 2 9. 4 4 4 + ( = 2 = $.5. 5 + 5 + 5 + 5 = 4.5 (each ticket is worth $4.5 on average.