Expected Value Calculator

Enter each possible outcome with its value and probability, and the calculator finds the expected value: the average result per trial if the situation were repeated many times. It also gives the variance and standard deviation, which measure how spread out the results are, and the chance of a positive result. Probabilities can be typed as decimals, percentages or fractions. A chart shows the probability of each outcome. Use it for games of chance, insurance, investment scenarios and business decisions.

E(X) = Σ x × P(x)

Var(X) = Σ P(x) × (x − E(X))²    SD = √Var(X)

Multiply each value by its probability and add the results. The probabilities of all outcomes must add up to exactly 1; if they do not, an outcome is missing or a probability is wrong, and the calculator says so rather than guessing.

A fair die. Each face from 1 to 6 has a one-in-six chance, so the expected value is 3.5 and the standard deviation 1.71. You can never roll 3.5; the expected value is a long-run average, not a likely single result.

Roulette. A $1 bet on a single number in American roulette pays $35 with a 1 in 38 chance and loses $1 otherwise. The expected value is -0.0526 dollars, a loss of about 5.3 cents per dollar bet over time, which is the casino's edge.

A business decision. A product launch with a 30% chance of making $200,000, 50% chance of $50,000 and 20% chance of losing $80,000 has an expected value of $69,000, but a standard deviation of $98,838 shows how uncertain that is.

Two choices with the same expected value can carry very different risk. A guaranteed $50 and a coin flip for $100 or nothing both have an expected value of $50, but most people prefer the certain amount. That preference is why insurance makes sense even though its expected value is negative for the buyer: it trades a small certain loss for protection against a large, unlikely one. The standard deviation helps show how much a decision could swing, and the chance of a gain shows how often it pays off.

For a lottery ticket, list each prize minus the ticket price with its probability, and the loss of the ticket for everything else. Expected values are almost always negative, because operators pay out less than they collect. Large jackpots can raise the expected value of a ticket, but the chance of sharing the prize with other winners and taxes on winnings bring it back down. The lottery number generator shows the exact odds for common games.

Expected value is the basis of decision analysis. For each option, list the outcomes, estimate their probabilities and values, and compare the expected values. The hardest part is usually the probabilities, so try a range of estimates and see whether the best choice changes; if it does not, the decision is robust. Including the cost of being wrong, not just the average, leads to choices you can live with when luck goes against you.

• Use net values: subtract what it costs to take part from each payoff.
• Make the outcomes cover every possibility, with no overlaps.
• Fractions typed as 1/38 are exact; rounded decimals can make the total slightly off 1.
• For repeated independent trials with two outcomes, such as counting successes, the binomial distribution calculator is the better tool.