Main Index > Expected value
Before you begin, you may want to read this overview: What is an Expected Value?
If you make a chart, the math behind finding an expected value becomes clearer. 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 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 in probability and statistics 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.
Back to Top