Section PC.3 – Expected Value
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 1 |
| In the casino game roulette, a wheel with 38 spaces (18 red, 18 black, and 2 green) is spun. In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?
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| Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you’ve lost $2. On a per-space basis, you have “won” -$2/$38 ≈ -$0.053. In other words, on average you lose 5.3 cents per space you bet on. |
We call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).
| Example 2 | ||||||||||||
| Refer back to the roulette example. The wheel that is spun has 38 spaces (18 red, 18 black, and 2 green). In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?
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| There is another way to compute expected value without imagining what would happen if we play every possible space. There are 38 possible outcomes when the wheel spins, so the probability of winning is [latex]\frac{1}{38}[/latex] The complement, the probability of losing, is [latex]\frac{37}{38}[/latex].
Notice that if we multiply each outcome by its corresponding probability we get $35 · [latex]\frac{1}{38}[/latex] = 0.9211 and −$1 · [latex]\frac{37}{38}[/latex] = −0.9737 , and if we add these numbers we get 0.9211 + (-0.9737) ≈ -0.053, which is the expected value of the game. Note: We use $35 (not $36) because you had to pay $1 in order to win $36. When you win you actually gain $35.
If we play roulette many times, we can expect that on average we will lose $0.053 per play. This expected value is negative, which indicates that on average we will lose money. |
| Expected Value |
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Expected Value is the average gain or loss of an event if the procedure is repeated many times. We can compute the expected value by multiplying every possible outcome by the probability of that outcome, then adding up the products. |
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.
| You Try PC.3.A |
| A raffle ticket costs $3. There is 1 winning ticket out of the 150 tickets sold. The winner gets $300. What is the expected value (to you) of the raffle ticket? Should you buy a ticket? |
Expected value also has applications outside of gambling. Expected value is very common in making insurance decisions.
| Example 3 | ||||||||||||
| A 40-year-old man in the U.S. has a 0.242% risk of dying during the next year. 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? | ||||||||||||
The probabilities and outcomes are:
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. |
| Example 4 | ||||||||||||
| Consider the same problem we just did, but we will answer it using a slightly different setup.
A 40-year-old man in the U.S. has a 0.242% risk of dying during the next year. 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? |
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| The man will pay $275 for the policy whether or not he dies in the next year, so the probability of losing the $275 is 100%.
The probabilities and outcomes are:
The expected value is (-$275)(1) + ($100,000)(0.00242) + = -$33. |
| Example 5 |
| A company estimates that 0.3% of their products will fail after the original warranty period but within 2 years of the purchase, with a replacement cost of $500. If they offer a 2 year extended warranty for $40, what is the company’s expected value of each warranty sold? |
| The company will receive the $40 for the warranty whether or not the product fails, so the probability of receiving the $40 is 100%. The company has a 0.3% chance of having to pay the $500 replacement cost if the product fails.
The expected value (to the company) is $40(1) + (-$500)(0.003) = $38.50 On average, the company will earn $38.50 for each policy sold. |
Section PC.3 Answers to You Try Problems
PC.3.A
The expected value of the raffle ticket is -$1. You should expect to lose $1 for each ticket that you buy. You shouldn’t buy a ticket.
