Expected value is exactly what you might think it means intuitively: the return you can expect for some kind of action, like how many questions you might get right if you guess on a multiple choice test.
For example, if you take a 20 question multiple-choice test with A,B,C,D as the answers, and you guess all “A”, then you can expect to get 25% right (5 out of 20). The math behind this kind of expected value is:
The probability (P) of getting a question right if you guess: .25
The number of questions on the test (n)*: 20
P x n = .25 x 20 = 5
*You might see this as X instead.
This type of expected value is called an expected value for a binomial random variable. It’s binomial because there are only two possible outcomes: you get the answer right, or you get the answer wrong.
The expected value formula for binomial random variables is written as E(X)=n*P (or P*n, X*P or P*X).
The expected value formula changes a little if you have a series of trials (for example, a series of coin tosses). When you have a series of trials, you take your basic formula (n*P) for each trial and then add them together. Mathematically, the expected value formula for a series of binomial trials is:
E(X) = x1P1 + x2P2 + x3P3 + . . . + xnPn.
The “…+ xnPn” just means to keep on adding your results together — even if you have 1,000 results you would still keep on adding.
If you make a chart, the math behind finding an expected value becomes clearer.
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This article explains how to figure out the expected value for a single item (like purchasing a single raffle ticket) and what to do if you have multiple items. If you have a discrete random variable, read this other article instead: Expected value for a discrete random variable.
Sample question: You buy one $10 raffle ticket for a new car valued at $15,000. Two thousand tickets are sold. What is the expected value of your gain?
Step 1: Make a probability chart (see: How to construct a probability distribution). Put Gain(X) and Probability P(X) heading the rows and Win/Lose heading the columns.
Step 2: Figure out how much you could gain and lose. In our example, if we won, we’d be up $15,000 (less the $10 cost of the raffle ticket). If you lose, you’d be down $10. Fill in the data (I’m using Excel here, so the negative amounts are showing in red).
Step 3: In the bottom row, put your odds of winning or losing. Seeing as 2,000 tickets were sold, you have a 1/2000 chance of winning. And you also have a 1,999/2,000 probability chance of losing.
Step 4: Multiply the gains (X) in the top row by the Probabilities (P) in the bottom row.
$14,990 * 1/2000 = $7.495,
Step 5:Add the two values together:
$7.495 + -$9.995 = -$2.5.
Note on multiple items: for example, what if you purchase a $10 ticket, 200 tickets are sold, and as well as a car, you have runner up prizes of a CD player and luggage set?
Perform the steps exactly as above. Make a probability chart except you’ll have more items:
Then multiply/add the probababilities as in step 4: 14,990*(1/200) + 100 * (1/200) + 200 * (1/200) + -$10 * (197/200).
You’ll note now that because you have 3 prizes, you have 3 chances of winning, so your chance of losing decreases to 197/200.
Note on the formula: The actual formula for expected gain is E(X)=∑X*P(X) (this is also one of the AP Statistics formulas). What this is saying (in English) is “The expected value is the sum of all the gains multiplied by their individual probabilities.”
Use our online expected value calculator to calculate simple probabilities!
Step 1: Type your values into two columns in Excel (“x” in one column and “f(x)” in the next.
Step 2: Click an empty cell.
Step 3: Type =SUMPRODUCT(A2:A6,B2:B6) into the cell where A2:A6 is the actual location of your x variables and f(x) is the actual location of your f(x) variables.
Step 4: Press Enter.
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Expected values for binomial random variables are probably the simplest type of expected values. In real life, you’re likely to encounter more complex expected values that have more than two possibilities. For example, you might buy a scratch off lottery ticket with prizes of $1000, $10 and $1.
Online expected value calculator
If you’re looking for more information on formula variations (this gets a bit more technical!), see this article at Wolfram.