Expected Value Calculator: weigh a bet or decision by its long-run average

Expected value is the average result of a wager or decision if it could be repeated many times. It is the single most useful number for judging whether a bet, a game or a business choice is worth taking, because it folds the chance of each outcome and the size of each payoff into one figure. A positive expected value means the decision pays over the long run. A negative expected value means it costs over the long run, however the next single attempt turns out. This calculator computes expected value from the probability of winning, the amount you stand to win, the probability of losing and the amount you stand to lose.

What expected value means

Expected value is a weighted average. Each possible outcome contributes its probability multiplied by its value, and the total is the expected value. If a fair coin pays you 2 units for heads and takes 1 unit for tails, the expected value is 0.5 times 2 plus 0.5 times negative 1, which is 0.5 units per flip. Over a thousand flips you would expect to be up about 500 units, even though any single flip is a coin toss.

The word expected is a technical term and does not mean the result you personally expect this time. An event with a negative expected value can still win in the short run, and a positive expected value decision can still lose. The number describes the centre of the distribution over many repetitions, which is exactly what matters when the same choice is made over and over.

The formula

The expected value of a bet with two outcomes is:

expected value = (probability of winning times the net win) minus (probability of losing times the stake)

The net win is what you receive beyond your stake. If you stake 10 units at decimal odds of 2.50, a win returns 25 units in total, so the net win is 15 units. The probability of losing is one minus the probability of winning, because the two must add to one. The full expression is then:

EV = P(win) x net win - (1 - P(win)) x stake

The result is in the same units as the stake, so a value of 0.10 means a long-run gain of one tenth of a unit per bet, and a value of negative 0.10 means a long-run loss of one tenth of a unit per bet.

Where the formula comes from

The two-outcome form is a special case of the general definition. Expected value is the sum over every outcome of the probability of that outcome times its payoff. With only win and lose as outcomes, the losing branch has probability one minus the winning probability, and its payoff is the stake lost, which is negative. Grouping the two terms gives the familiar win and lose form above.

The same structure appears far beyond betting. Insurance pricing, casino game design, product decisions and investment sizing all rest on expected value. A casino does not need to win every spin, only to hold a small edge per spin that compounds over millions of spins. The edge is simply the expected value per unit staked, with the sign flipped.

Worked examples

Take a coin flip offered at decimal odds of 1.90. The probability of winning is 0.5 and the probability of losing is 0.5. On a 1 unit stake, a win returns 1.90 units, so the net win is 0.90 units. The expected value is 0.5 times 0.90 minus 0.5 times 1.00, which is 0.45 minus 0.50, or negative 0.05 units per flip. That five hundredths of a unit is the house edge, and over 100 flips it accumulates to an expected loss of 5 units.

Now take the same coin flip at a fair price. Decimal odds of 2.00 on a 1 unit stake give a net win of 1.00. The expected value is 0.5 times 1.00 minus 0.5 times 1.00, which is zero. A fair coin at fair odds has zero expected value, which is the definition of a fair bet.

A genuinely positive example needs an edge in your favour. If you believe an event priced at 3.00 decimal actually happens 40 percent of the time, then on a 10 unit stake the net win is 20 units. The expected value is 0.4 times 20 minus 0.6 times 10, which is 8 minus 6, or positive 2 units per bet. The bet is positive expectation only because your estimate of 40 percent is higher than the market's implied 33.33 percent. If the market is right and the event happens 33.33 percent of the time, the same bet has an expected value of 0.3333 times 20 minus 0.6667 times 10, which is negative 0.0 units, a break-even after rounding.

Reading the result

A positive expected value is a signal to take the bet, on one condition: your estimate of the true probability must be sound. The calculator cannot know the true probability, only the one you enter, so the quality of the answer is the quality of your estimate. A negative expected value is a signal the bet is priced against you, which is the normal state of most bookmaker and casino odds.

Expected value also helps with sizing. A bet can have positive expected value and still be too large for your bankroll, because a run of losses can wipe you out before the long run arrives. Expected value says whether the bet is good on average; a separate sizing decision, such as a fixed fraction of your bankroll, decides how much to commit. The two questions are different and both matter.

Keep the time horizon honest. Expected value is a long-run average. One bet is one draw from a noisy distribution, and the noise can be large. A positive expected value is an edge, not a guarantee, and a negative expected value is a tax, not a certainty of immediate loss. Over a single attempt, luck dominates. Over many attempts, the edge dominates.

Expected value in everyday decisions

The same calculation appears in choices that have nothing to do with gambling. A subscription that costs 10 units and returns value you judge at 20 units with certainty has an expected value of 10 units, so it is worth taking. A stock you estimate has a 60 percent chance of rising 5 units and a 40 percent chance of falling 5 units has an expected value of 0.6 times 5 minus 0.4 times 5, which is 1 unit, a small but positive edge. The arithmetic is identical to the betting case: list the outcomes, weight each by its probability, and add them up.

The discipline comes from being honest about the probabilities. People routinely overestimate the chance of a good outcome and underestimate the chance of a bad one, which flips a negative expected value decision into a positive one on paper. The calculator only carries out the arithmetic; the judgement that feeds it is the hard part.

Frequently Asked Questions

What does a negative expected value mean? It means the bet loses money on average over many repetitions, even if it can win on any single attempt. Most casino games and bookmaker odds carry a small negative expected value for the player, which is how the house and the bookmaker earn.

Can a bet with positive expected value still lose? Yes. Expected value is an average over many trials, not the result of one trial. A positive expected value bet loses whenever the unlikely losing outcome happens, which it will do from time to time. The edge only shows up as the number of trials grows.

How is expected value different from probability? Probability says how likely a win is. Expected value says how much the bet is worth per attempt, combining the probability with the size of the win and the loss. A bet can have a high probability of winning and still have negative expected value if the occasional loss is large, which is why both numbers matter.

What is the house edge? The house edge is the negative expected value per unit staked, expressed as a percentage, but viewed from the operator's side it is positive. A bet with an expected value of negative 0.05 units on a 1 unit stake has a house edge of 5 percent. The player's expected value and the house edge are the same number with opposite signs.

Do I need the exact probability to use this calculator? You need an estimate, and the result is only as good as that estimate. Where the true probability is known, as with a fair coin or a die, the expected value is exact. Where the probability is your judgement, such as a sports event, the expected value is a guide that should be treated with the same caution as the estimate behind it.

How does expected value help with a run of losses? It does not prevent a run of losses, but it tells you whether to keep playing through one. If the expected value is positive, the long run favours you and a losing streak is noise. If it is negative, the long run is against you and no streak management changes that fact.

Is a break-even expected value worth taking? A zero expected value decision neither helps nor hurts over the long run, so it is a matter of preference rather than arithmetic. It can still be worth taking for enjoyment, but it should never be mistaken for an edge.