How to use the expected value calculator
- Enter each possible outcome on its own row with a value, such as a prize or a loss, and the probability of that outcome. Use a minus sign for a loss.
- Choose whether you are typing the probabilities as percents or as decimals. They must add up to exactly 100% (or 1).
- Open the advanced option if there is a cost to play, such as a ticket price, to see the expected value after that cost.
- Read the expected value, the variance and the standard deviation. The table shows how much each outcome contributes.
Formula
Expected value = sum of (value x probability)
Variance = sum of probability x (value - expected value)^2
Standard deviation = square root of the variance
Expected value is the long-run average per play. It is not what you should expect to see on any one play.
A raffle ticket
A ticket wins $10 with probability 20% and wins nothing with probability 80%. The expected value is 10 x 0.20 + 0 x 0.80 = $2. The variance is 0.20 x (10 - 2)^2 + 0.80 x (0 - 2)^2 = 12.8 + 3.2 = 16, so the standard deviation is 4.
Now take a game with a $100 prize at 1%, a $50 prize at 4% and nothing at 95%. The expected value is 1 + 2 + 0 = $3. If a ticket costs $2, the expected value after the cost is $1.
What expected value means
If you could repeat the same uncertain choice many times, the average result would settle near the expected value. A game with an expected value of $2 per play pays about $2 a play over a long run, even though no single play pays exactly that.
Two choices with the same expected value can be very different. A sure $2 and a 20 percent chance at $10 have the same expected value, but the second has a standard deviation of 4 and the first has none. The spread tells you how bumpy the ride is.
Using it for decisions
Expected value is a good first step for comparing options that you can repeat, such as a pricing test or a recurring bet. It is less useful for one-off, high-stakes choices, where how much you can afford to lose matters as much as the average.
If your probabilities come from a test or a rate, the probability calculator, the binomial probability calculator and the Bayes theorem calculator can help you work them out first.
Assumptions and limits
- The outcomes are the ones you list, and the probabilities add up to exactly 100 percent. The tool refuses to compute otherwise.
- Each play is independent and the probabilities do not change from play to play.
- Values are in the same unit, such as dollars, and a cost to play is subtracted once per play.
- Taxes, fees and the time value of money are not modeled.
Frequently asked questions
How do you calculate expected value?
Multiply each outcome by its probability and add the results. An outcome of $10 at 20 percent and $0 at 80 percent gives an expected value of $2.
Why must the probabilities add up to 100 percent?
Because the list is meant to cover every possible result. If they add up to less, an outcome is missing, and if they add up to more, some chance is counted twice. The tool shows the total you entered so you can fix it.
Can expected value be negative?
Yes. If the average result per play is a loss, the expected value is negative. Many games with a cost to play have a negative expected value after the cost is subtracted.
Does a positive expected value mean I will win?
No. It describes the average over many repeats. A single play can still lose, and the standard deviation shows how large the swings can be.