Expected Value Calculator
Expected value is the long run average of a gamble. Weight every outcome by how likely it is, add the results, and you have the number that decides whether a bet is worth taking.
Expected value
43
- Variance of the outcomes
- 1,691
- Standard deviation
- 41.1218
- Total of the probabilities
- 100.00%
- Best outcome
- 100
- Worst outcome
- -20
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
The formula this calculator uses
E = p1 x x1 + p2 x2 + ... + pn x xn
- E
- The expected value, a long run average
- xi
- Each possible outcome, profit or loss
- pi
- The probability of that outcome, as a decimal
- sum of pi
- Must equal one, or the inputs are incomplete
How to check the result by hand
- 1
List every outcome with its probability
Write down each result and how likely it is. Outcomes of 100, 50, 0 and minus 20 with probabilities of 25, 40, 25 and 10 percent cover the whole space.
- 2
Check the probabilities add to one
25 plus 40 plus 25 plus 10 is 100 percent. If the total falls short, an outcome is missing and the expected value will be wrong by exactly the unlisted part.
- 3
Multiply each outcome by its probability
100 times 0.25 is 25, 50 times 0.40 is 20, 0 times 0.25 is 0, and minus 20 times 0.10 is minus 2. These are each outcome's contribution.
- 4
Add the contributions
25 plus 20 plus 0 minus 2 is 43, the expected value. Compare it against the spread of 41.1218 to see how much of the story the average leaves out.
For worked examples, common mistakes and the limits of this formula, read the full How To Calculate Expected Value page.