Expected Value Calculator

What expected value measures

Expected value answers one question in plain money terms: if you ran this bet over and over, what would your average result be per try? It is not a prediction for any single flip, ticket or hand. It is the long-run average, and it is the only number that separates a bet worth repeating from one that quietly drains a wallet.

A positive expected value means repeating the bet tends to grow your money. A negative one means repetition grinds it down, no matter how good a lucky night feels. Statisticians treat this average as the balance point of a distribution, the same way a data set's mean is its center of mass. The NIST/SEMATECH Engineering Statistics Handbook describes the mean as the typical value a distribution balances on, which is exactly what a per-try average becomes once every outcome is weighted by its chance.

The idea reaches well past casinos. An extended warranty, a raffle ticket, a client who might not pay, a repair-or-replace call: each is a set of outcomes with chances attached, and each has an expected value you can put a dollar figure on before you commit.

The formula, worked line by line

The formula is one line: multiply the value of each outcome by its chance, then add every product up. Written out, it reads EV = (chance of A x value of A) + (chance of B x value of B), continuing for as many outcomes as the bet has.

Take the simplest bet there is. A friend offers a coin flip: heads wins you $2, tails costs you $1, and nothing is owed up front. Two outcomes, each with a chance of 50 percent, fill the table below.

The contributions add to 50 cents, so the expected value is $0.50 per flip. Run the same flip 100 times and the expected total is 100 x $0.50 = $50, though any single run of 100 can land far from that. Row order never matters, and an outcome with a 0 percent chance contributes nothing, so unused rows in the calculator can be left at zero.

One bookkeeping rule keeps the math honest: enter net values, meaning what your bankroll truly changes by. If a prize costs $5 to chase, the winning row is the prize minus $5 and the losing row is negative $5, not zero.

Reading the house edge in cents

Most real bets carry a negative expected value by design, and the size of that negative number is the price of playing. Casinos publish it as the house edge; the same reading works on any bet once you divide by the stake.

A single-number bet on an American roulette wheel makes the arithmetic concrete. The wheel has 38 pockets, a straight-up bet pays 35 to 1, and $1 rides on every spin.

The contributions add to -$0.05, which is 5.3 cents per spin. Read the other way, the bet returns about 94.7 cents per dollar staked, and the gap between that and a fair 100 cents is the edge. Over 1,000 spins of $1 each, the expected cost is about $52.60, which is why no betting pattern moves the average: the edge is baked into the chances and payouts, not the order of results.

Break-even chance: the number a bet has to beat

Expected value also tells you the chance a bet would need in order to be fair. For a bet with one prize and one losing outcome, the break-even chance equals the stake divided by the net prize, because that is the chance where chance x prize = stake exactly.

The town raffle in the next section shows the check in action. A $5 ticket chases a $1,500 prize, so the break-even chance is 5/1500 = 0.33 percent. The actual chance is 1 ticket in 500, which is 0.2 percent. The real chance sits below break-even, so the ticket must lose on average, and the calculator confirms it: 0.2 percent x $1,495 plus 99.8 percent x -$5 = -$2.00 per ticket.

The same test runs in reverse. If a promotion offered a 1 percent shot at a $500 prize for $4, the break-even chance would be 4/500 = 0.8 percent, and the offered 1 percent beats it. The expected value works out to +$1.00 per try. Offers with a positive edge are rare, which is exactly why the thirty-second check is worth running whenever a prize looks tempting.

Three everyday decisions, worked end to end

The same rows solve decisions that never touch a casino. Each case below enters net values, sums its chances to 100 percent, and reads the verdict straight off the expected value.

The raffle. 500 tickets at $5 each, one $1,500 prize. Win row: 0.2 percent at +$1,495. Lose row: 99.8 percent at -$5. EV = -$2.00 per ticket, so a booth that sells all 500 keeps $1,000 and the average player donates $2 a ticket.

The extended warranty. A $120 plan against an 8 percent chance of a $900 repair. Repair row: 8 percent at +$780 (the $900 payout minus the $120 premium). No-repair row: 92 percent at -$120. EV = 0.08 x $780 + 0.92 x -$120 = $62.40 - $110.40 = -$48.00. The plan loses money on average, which is normal for insurance; buying it is a choice about absorbing a rare $900 hit, not a bet to win.

The side gig. A $40 job that needs $6 of fuel, with a 25 percent chance the client never pays. Paid row: 75 percent at +$34. Unpaid row: 25 percent at -$6. EV = 0.75 x $34 + 0.25 x -$6 = $25.50 - $1.50 = +$24.00 per job, or $480 across 20 such jobs.

Why a lucky night does not beat the math

Expected value is the center of the distribution, not a promise about any single run. The coin flip at the top of the page averages 50 cents per flip, but a single flip pays $2 or costs $1, so the spread around that average is wide: the standard deviation works out to $1.50 per flip, three times the average itself.

Volume shrinks the spread relative to the total. Over 100 flips the expected total is $50, and the spread of that total grows only as the square root of the number of flips, to about $15. Roughly two runs in three land within $15 of the expected $50, and the share landing below zero keeps shrinking as flips pile up. The same logic is why a casino with a 5-cent edge on the dollar does not care about one lucky player: across thousands of spins the law of large numbers pulls the realized average toward the edge with near certainty.

Short runs scatter, and that is the trap. A raffle buyer who wins once remembers the $1,495 prize and not the losing tickets that paid for it. The expected value line in the calculator is the honest summary: it already priced in every losing outcome at its own chance.

Setting up rows for your own bet

Three habits keep the input honest. First, use net values in every row: what your bankroll truly changes by, with the cost of playing already subtracted. Second, make the chances sum to 100 percent; the results panel shows the running total and flags anything else. Third, set unused rows to 0 percent chance so they contribute nothing.

Converting the chance formats people meet in the wild:

Odds written "7 to 2 against" mean 2 wins in 9 tries, and decimal odds convert as 1 divided by the odds figure. If the rows refuse to sum to 100, one outcome is usually missing: a push, a refund, or the chance that nothing happens at all.

Where expected value stops helping

A negative expected value is not always a mistake. Insurance of every kind runs at negative EV by design: the $48 average loss on the warranty above is the price of not eating a $900 repair alone. The math cannot price that peace of mind, so treat insurance EV as a cost quote, not a verdict.

Missing outcomes are the classic input error. Leaving out a 5 percent chance of a $5,000 loss changes the answer by $250 per try, more than most bets are worth. Before trusting a result, walk through every way the situation can end and give each one a row.

Taxes push real results lower than the math shows. The IRS taxes gambling winnings in full, and losses are deductible only up to winnings and only if you itemize (see IRS Topic 419). A $1,000 win taxed at 22 percent leaves $780, while a $1,000 losing ticket offsets nothing unless you itemize and have winnings to match.

And the number says nothing about control. If betting stops feeling like entertainment, free and confidential help is one call away: the National Council on Problem Gambling helpline at 1-800-MY-RESET puts trained counselors on the line.