# 9.8: Expected Value

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Expected value is perhaps the most useful probability concept we will discuss. It has many applications, from insurance policies to making financial decisions, and it's one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.

## Example 42

###### Solution

Earlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers:

$$\frac{_6 C_{6}}{_{48} C_{6}}=\frac{1}{12271512} \approx 0.0000000815$$ for all 6 numbers,

$$\frac{\left(_{6} C_{5}\right)\left(_{42} C_{1}\right)}{_{48} C_{6}}=\frac{252}{12271512} \approx 0.0000205$$ for 5 numbers.

Our probabilities and outcome values are:

$$\begin{array}{|l|l|} \hline \text { Outcome } & \text { Probability of outcome } \\ \hline \ 999,999 & \frac{1}{12271512} \\ \hline \ 999 & \frac{252}{12271512} \\ \hline-\ 1 & 1-\frac{253}{12271512}=\frac{12271259}{12271512} \\ \hline \end{array}$$

The expected value, then is:

$$(\ 999,999) \cdot \frac{1}{12271512}+(\ 999) \cdot \frac{252}{12271512}+(-\ 1) \cdot \frac{12271259}{12271512} \approx-\ 0.898$$

On average, one can expect to lose about 90 cents on a lottery ticket. Of course, most players will lose $1. In general, if the expected value of a game is negative, it is not a good idea to play the game, since on average you will lose money. It would be better to play a game with a positive expected value (good luck trying to find one!), although keep in mind that even if the average winnings are positive it could be the case that most people lose money and one very fortunate individual wins a great deal of money. If the expected value of a game is 0, we call it a fair game, since neither side has an advantage. Not surprisingly, the expected value for casino games is negative for the player, which is positive for the casino. It must be positive or they would go out of business. Players just need to keep in mind that when they play a game repeatedly, their expected value is negative. That is fine so long as you enjoy playing the game and think it is worth the cost. But it would be wrong to expect to come out ahead. ##### Try it Now 13 A friend offers to play a game, in which you roll 3 standard 6-sided dice. If all the dice roll different values, you give him$1. If any two dice match values, you get $2. What is the expected value of this game? Would you play? Answer Suppose you roll the first die. The probability the second will be different is $$\dfrac{5}{6}$$. The probability that the third roll is different than the previous two is $$\dfrac{4}{6}$$, so the probability that the three dice are different is $$\dfrac{5}{6} \cdot \dfrac{4}{6}=\dfrac{20}{36}$$. The probability that two dice will match is the complement, $$1-\dfrac{20}{36}=\dfrac{16}{36}$$. The expected value is: $$(\ 2) \cdot \dfrac{16}{36}+(-\ 1) \cdot \dfrac{20}{36}=\dfrac{12}{36} \approx \ 0.33$$. Yes, it is in your advantage to play. On average, you’d win$0.33 per play.

Expected value also has applications outside of gambling. Expected value is very common in making insurance decisions.

## Example 44

A 40-year-old man in the U.S. has a 0.242% risk of dying during the next year[2]. An insurance company charges $275 for a life-insurance policy that pays a$100,000 death benefit. What is the expected value for the person buying the insurance?

###### Solution

The probabilities and outcomes are

$$\begin{array}{|l|l|} \hline \text { Outcome } & \text { Probability of outcome } \\ \hline \ 100,000-\ 275=\ 99,725 & 0.00242 \\ \hline-\ 275 & 1-0.00242=0.99758 \\ \hline \end{array}$$

The expected value is $$(\ 99,725)(0.00242)+(-\ 275)(0.99758)=-\ 33$$.

Not surprisingly, the expected value is negative; the insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people.

For people buying the insurance, there is a negative expected value, but there is a security that comes from insurance that is worth that cost.

[1] Photo CC-BY-SA http://www.flickr.com/photos/stoneflower/

[2] According to the estimator at www.numericalexample.com/inde...=article&id=91

This page titled 9.8: Expected Value is shared under a CC BY-SA license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) .