Expected Value Calculator
Add outcome values and their probabilities to compute E[X] = the sum of value times probability, with a per-term breakdown.
Expected value calculator
Expected value, written E[X], is the long-run average outcome of a random situation — the number you would converge to if you repeated the situation many, many times. Enter each possible outcome's value and its probability, and this calculator multiplies each value by its probability, adds the results, and shows the full per-term breakdown. It also checks that your probabilities add up to about 1, the requirement for a complete distribution.
The formula
Expected value is a probability-weighted average of every possible outcome:
E[X] = Σ (value · probability) = x₁p₁ + x₂p₂ + … + xnpn
Each outcome contributes its value scaled by how likely it is, so rare outcomes pull the average only a little and common ones pull it a lot. Because the probabilities of a complete distribution sum to 1, the expected value is genuinely a weighted average rather than a plain total.
Expected value in gambling and decisions
Expected value is the core tool for judging bets and decisions under uncertainty. A bet is "fair" when its expected value is zero, profitable when it is positive, and a long-run loss when it is negative. Almost every casino game has a negative expected value for the player by design — that built-in edge is the house advantage. In business and everyday decisions, comparing the expected values of your options is a rational way to choose when outcomes are uncertain, though you should still weigh risk and the size of the worst case, not expected value alone.
Worked examples
A simple gamble pays +10 with probability 0.5 and −4 with probability 0.5. Its expected value is (10 × 0.5) + (−4 × 0.5) = 5 − 2 = +3, a favorable bet. By contrast, a $1 straight-up roulette bet pays +35 with probability 1/38 and −1 with probability 37/38, for an expected value of about −$0.05 — you lose about a nickel per dollar wagered on average. To explore the random processes behind these probabilities, see the binomial distribution calculator. Everything is computed locally in your browser.
Expected value calculator
Expected value, written E[X], is the long-run average outcome of a random situation — the number you would converge to if you repeated the situation many, many times. Enter each possible outcome's value and its probability, and this calculator multiplies each value by its probability, adds the results, and shows the full per-term breakdown. It also checks that your probabilities add up to about 1, the requirement for a complete distribution.
The formula
Expected value is a probability-weighted average of every possible outcome:
E[X] = Σ (value · probability) = x₁p₁ + x₂p₂ + … + xnpn
Each outcome contributes its value scaled by how likely it is, so rare outcomes pull the average only a little and common ones pull it a lot. Because the probabilities of a complete distribution sum to 1, the expected value is genuinely a weighted average rather than a plain total.
Expected value in gambling and decisions
Expected value is the core tool for judging bets and decisions under uncertainty. A bet is "fair" when its expected value is zero, profitable when it is positive, and a long-run loss when it is negative. Almost every casino game has a negative expected value for the player by design — that built-in edge is the house advantage. In business and everyday decisions, comparing the expected values of your options is a rational way to choose when outcomes are uncertain, though you should still weigh risk and the size of the worst case, not expected value alone.
Worked examples
A simple gamble pays +10 with probability 0.5 and −4 with probability 0.5. Its expected value is (10 × 0.5) + (−4 × 0.5) = 5 − 2 = +3, a favorable bet. By contrast, a $1 straight-up roulette bet pays +35 with probability 1/38 and −1 with probability 37/38, for an expected value of about −$0.05 — you lose about a nickel per dollar wagered on average. To explore the random processes behind these probabilities, see the binomial distribution calculator. Everything is computed locally in your browser.
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Frequently Asked Questions
Common questions about the Expected Value Calculator
Expected value, E[X], is the long-run average outcome of a random process — the value you would average if you repeated the process many times. It is calculated by multiplying each possible outcome by its probability and adding the results, making it a probability-weighted average of all outcomes.
Multiply each outcome's value by its probability, then sum across all outcomes: E[X] = Σ (value × probability). For a bet that pays +10 half the time and −4 the other half, E[X] = (10 × 0.5) + (−4 × 0.5) = 3. This calculator does the multiplication and summation and shows each term.
For a complete probability distribution, yes — the probabilities of all possible outcomes should sum to 1 (or 100%). This calculator warns you when they do not, because a missing or mistyped probability makes the expected value inaccurate. Each probability should also be between 0 and 1.
A negative expected value means that, on average, you lose over many repetitions. Most casino games have a negative expected value for the player — that is the house edge. A positive expected value means you gain on average, and an expected value of zero describes a fair game with no long-run advantage to either side.
No. Expected value is a weighted average and may not even be a possible outcome — the expected value of a single fair die roll is 3.5, which never actually appears. The most likely outcome is the one with the highest probability. The two can differ sharply, especially in skewed distributions.
ℹ️ Disclaimer
This tool is provided for informational and educational purposes only. All processing happens entirely in your browser - no data is sent to or stored on our servers. While we strive for accuracy, we make no warranties about the completeness or reliability of results. Use at your own discretion.