Expected Value
Learning Outcomes
- Compute a conditional probability for an event
- Use Baye’s theorem to compute a conditional probability
- Calculate the expected value of an event
Repeating Procedures Over Time
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
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?Answer: 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). 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]. Summarizing these along with the values, we get this table:
Outcome | Probability of outcome |
$35 | [latex]\frac{1}{38}[/latex] |
-$1 | [latex]\frac{37}{38}[/latex] |
Expected Value
- Expected Value is the average gain or loss of an event if the procedure is repeated many times.
Try It
You purchase a raffle ticket to help out a charity. The raffle ticket costs $5. The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth $4000. Compute the expected value for this raffle.Example
In a certain state's lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. If they match 5 numbers, then win $1,000. It costs $1 to buy a ticket. Find the expected value.Answer: Earlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers: [latex]\frac{{}_{6}{{C}_{6}}}{{}_{48}{{C}_{6}}}=\frac{1}{12271512}\approx0.0000000815[/latex] for all 6 numbers, [latex]\frac{\left({}_{6}{{C}_{5}}\right)\left({}_{42}{{C}_{1}}\right)}{{}_{48}{{C}_{6}}}=\frac{252}{12271512}\approx0.0000205[/latex] for 5 numbers. Our probabilities and outcome values are:
Outcome | Probability of outcome |
$999,999 | [latex]\frac{1}{12271512}[/latex] |
$999 | [latex]\frac{252}{12271512}[/latex] |
-$1 | [latex]1-\frac{253}{12271512}=\frac{12271259}{12271512}[/latex] |
Try It
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?Example
A 40-year-old man in the U.S. has a 0.242% risk of dying during the next year.[footnote]According to the estimator at http://www.numericalexample.com/index.php?view=article&id=91[/footnote] 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?Answer: The probabilities and outcomes are
Outcome | Probability of outcome |
$100,000 - $275 = $99,725 | 0.00242 |
-$275 | 1 – 0.00242 = 0.99758 |
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Expected Value. Authored by: David Lippman. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike.
- Roulette. Authored by: Chris Yiu. License: CC BY-SA: Attribution-ShareAlike.
- Expected value. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Expected value of insurance. Authored by: OCLPhase2's channel. License: CC BY: Attribution.