SigmaCalc

Expected Value Calculator

What you'd average per play if you repeated this bet or decision many, many times.

value:probability pairs (comma-separated)

Example: 100:0.2, 50:0.3, -20:0.5

Expected value

What you entered

Outcomes: 3

1
Outcome 1 contribution
100 × 0.2= 20
2
Outcome 2 contribution
50 × 0.3= 15
3
Outcome 3 contribution
-20 × 0.5= -10
4
Sum all contributions
20 + 15 + -10= 25

Result

Expected value: 25

Weighting each outcome by its probability and summing gives an expected value of 25 — the long-run average result if this were repeated many times.

Expected value of common scenarios

Expected value multiplies each outcome by its probability and sums the results - the long-run average per repetition.

Outcomes	Expected value
+10 at 50%, -5 at 50%	2.500
+100 at 10%, 0 at 90%	10.000
A fair six-sided die	3.500

The die row shows the important caveat: the expected value is 3.5, which is not an outcome the die can ever produce. Expected value is a long-run average, not a prediction of any single trial, and it says nothing about variance - the first two rows have very different risk profiles despite both being positive. That is why a positive expected value is not automatically a good bet if the downside would ruin you, and why insurance is rational to buy despite having negative expected value for the buyer. Probabilities across all outcomes must sum to 1.

Expected value isn't a likely single outcome

For a bet with a -$5 or +$50 payoff, the expected value might be a number like $2.50 — an amount that never actually occurs on any single play. It only describes the long-run average across many repetitions.

Where it's used

Expected value underlies decision theory, gambling odds, insurance pricing, and investment analysis — anywhere you need to compare options with uncertain, probability-weighted outcomes on a single common scale.

Frequently asked questions

A lottery ticket costs $2. There's a 1-in-1000 chance of winning $500. Is it worth buying?

EV = (1/1000 × $500) + (999/1000 × −$2) = $0.50 − $1.998 = −$1.50. On average, you lose $1.50 per ticket. The expected value is negative — mathematically, you shouldn't buy it. But people play lotteries for entertainment value, not expected value.

How is expected value different from the most likely outcome?

The expected value is the long-run average, which may not be a possible single outcome. A die roll has EV = 3.5, but you'll never roll a 3.5. The most likely outcome (mode) of a die is all faces equally at 1/6 each. EV is about averages; mode is about what's most common.

How do insurance companies use expected value?

They calculate EV of claims (probability of accident × average claim cost) and set premiums above that. If 1% of drivers have an accident costing $20,000 on average, EV = $200/year. They charge, say, $350 to cover overhead and profit. The math works because they insure millions of people.