Expected Value Calculator

Enter your win rate, average win, and average loss and get the expected value of one repetition, after the costs most versions of this formula quietly leave out. We also show the win rate you need just to break even, and how long a run has to be before an edge is distinguishable from luck.

Repeated bet or trade Outcomes and probabilities

Win rate (%) (how often this comes out ahead)

Average win ($) (your typical winner)

Average loss ($) (a positive number please: we subtract it for you)

Cost per trade ($) (optional: spread, commission, fees)

Number of trades (optional: shows the range over a run this long)

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What expected value actually tells you

Expected value is the average result of one repetition, taken over a run long enough for luck to cancel out. That is the whole idea, and the two words people misread are "average" and "long". It is not a forecast of your next trade, your next hand, or your next product launch. Run the numbers on a bet that pays 10 at 40%, nothing at 35%, and costs you 20 at 25%, and the expected value is exactly −1 per try, a result that literally cannot happen on any single attempt. Our loot drop calculator makes the same point from the other end: at the "expected" number of attempts, more than a third of players still have nothing.

For a repeated bet or trade the formula collapses to something you can do in your head, and that simplicity is exactly why it gets quoted so often and applied so badly. Three things decide whether the number on the page becomes money in the account, and none of them are in the formula: what each repetition costs you, what win rate you would need if the formula came out to zero, and how many repetitions it takes before an edge is louder than the noise around it. This calculator adds all three.

The formula

EV = (win rate × average win) − (loss rate × average loss) − cost per trade

Break-even win rate = (average loss + cost) ÷ (average win + average loss)

Standard deviation of one trade = (average win + average loss) × √(win rate × loss rate)

The win rate is a decimal fraction (60% is 0.60) and the loss rate is whatever is left over. The average loss goes in as a positive number, because the formula already subtracts it. The break-even line comes from setting EV to zero and solving for the win rate, which is why the average win and average loss are the only sizes in it. The standard deviation is the closed form for a two-outcome gamble, and it is worth noticing that cost does not appear in it: costs move the average and leave the spread exactly where it was.

For the textbook version with more than two outcomes, expected value is the sum of every outcome multiplied by its own probability, and variance is the sum of every probability multiplied by the squared distance from that expected value. The second mode on this page does both, term by term.

Worked example

A 60% win rate, a $400 average win, a $250 average loss. The win side contributes 0.60 × $400 = $240.00 per trade and the loss side takes back 0.40 × $250 = $100.00, so the gross expected value is $140.00 per trade. That is the number the formula gives, and it is arithmetically correct.

Now add $15.00 of spread and commission per round turn. Costs come off every trade, winner or loser, so the answer becomes $125.00 per trade, and the win rate you need just to stand still rises from 38.5% to 40.8%. You are running 60%, which is 19.2 points of real edge, and that is a lot.

The honesty layer: a single trade here has a standard deviation of $650.00 × √(0.60 × 0.40) = $318.43, which is 2.5 times the $125.00 you expect to make. Over 100 trades the expected total is $12,500.00 with a standard deviation of $3,184.34, so about two runs in three land between $9,315.66 and $15,684.34, and about 19 in 20 land between $6,258.81 and $18,741.19. Under our sample size rule the edge separates from luck after just 26 trades.

Read that last number as a lie detector, not a compliment. An edge that proves itself in 26 trades is enormous, which tells you something about the inputs rather than about you. A believable retail edge, say a 55% win rate at one to one with $10.00 of costs, is worth $10.00 a trade against a break-even line of 52.5%, and the same rule asks for 1,584 trades before you could tell it from noise.

The break-even win rate is the number that reframes the question

Almost every argument about strategies is really an argument about win rate, and win rate on its own is meaningless. Feed the formula a zero and solve, and the whole question becomes one line: the win rate you need is your average loss divided by the sum of your average win and your average loss. That gives the table this calculator prints with every result, and it is the part worth screenshotting.

At one to one you need 50.0%, which is the intuition everyone starts with. Make your winners twice the size of your losers and the bar drops to 33.3%. Make them three times the size and it drops to 25.0%, so you can be wrong three times out of four and still make money. Go the other way, taking half the size of the loss on each winner, and you need 66.7% just to stand still. This is why a 90% win rate strategy can quietly lose money and a 35% one can print: the ratio, not the hit rate, sets the bar, and the hit rate only has to clear it.

Costs move that bar up, always. They go on the top of the fraction, so a cheap instrument and an expensive one with identical price action are genuinely different strategies. If you want the same logic applied to a business rather than a trade, our break-even calculator asks the same question about units and fixed costs.

Costs are not a detail, they are the margin

Here is the mechanism, and it is the reason the same strategy can be excellent at one size and hopeless at another. Costs are roughly fixed per repetition. Profit scales with the size of your target. Spread, commission, and slippage cost about the same whether you are aiming at ten points or a hundred, so they are a rounding error on big targets and the entire margin on small ones.

Take the example above and shrink both the win and the loss by a factor of ten, to a $40 average win and a $25 average loss. The ratio is unchanged, the win rate is unchanged, and the gross expected value is exactly one tenth of what it was: $14.00 per trade. Leave the round turn at $15.00 and the strategy now loses $1.00 per trade. The break-even win rate has climbed from 38.5% to 61.5%, and the 60% that looked like a strong edge is now one and a half points short. Nothing about the trading changed. The arithmetic did.

Variance, and why the long run is longer than you think

A positive expected value is a claim about a long run, and most people badly underestimate how long. One trade in the worked example has a standard deviation of $318.43 against an expected $125.00. That ratio, spread divided by edge, is the thing that sets how much evidence you need, and it usually runs from a few times to a few dozen times.

Two facts do the work. Over n independent repetitions the expected total grows with n, but the standard deviation of that total grows only with √n. Edge wins eventually, which is the good news, and it wins slowly, which is the rest of it. So the honest output for a run of trades is a range, not a total, which is what this calculator gives you.

For "how long is long enough" we use a stated rule rather than a vibe: the number of repetitions at which your expected total sits two standard deviations clear of zero. At that separation, a strategy with no edge at all would reach your result by luck only about 2% of the time. Setting n × EV = 2 × SD × √n and solving gives n = 4 × variance ÷ EV². It is a rough tool with a normal approximation inside it, not a formal hypothesis test, and it is deliberately simple so you can check it. The believable 55% strategy above needs 1,584 trades, which is why a strategy can be genuinely good and still be underwater after 40 trades, and why 40 trades is not an answer to anything.

What expected value does not tell you

How much to risk. This is the big one, and the formula is completely silent on it. Expected value is an average per repetition. It has no opinion about position size, and betting too large on a real edge is the classic way to be right and broke at the same time, because a bad run can take the account to zero long before the long run arrives. Risk of ruin and Kelly criterion sizing are the tools built for that question. This page does not compute them, and no expected value figure is a substitute for one.

Whether your inputs are real. Every number on this page is downstream of a win rate and two averages you supplied. Those come from a sample, and a small sample flatters. If your averages come from 30 trades, so does your answer.

Whether averages describe your losses. An "average loss" assumes losses have a typical size. One trade held through a gap can be worth fifty ordinary losers, and no average built from the other fifty saw it coming.

One piece of context for why the cost line and the sample size line here are not pedantry. Regulators require retail brokers in several jurisdictions to publish how many of their client accounts lose money. When ESMA introduced the rule in March 2018 it reported that national regulators' reviews found 74 to 89 percent of retail accounts typically lose money, and in the UK the FCA requires each firm to display its own current figure and recalculate it every three months, so you can read the number for any regulated broker off its own page. We mention it neutrally and for one reason: those accounts belong to people who already know the formula at the top of this page. Costs and sample size are where the arithmetic and the account statement usually part company.

Last, the same note our odds calculator ends on. This page prices arithmetic. It does not endorse any strategy, market, or platform, there are no links to any here, and a positive number in the result box is a property of the inputs you typed, not a promise about the future.

Sources

Where the numbers on this page come from. We go to the body that publishes the figure, not to another calculator. See how we verify.

ESMA agrees to prohibit binary options and restrict CFDs to protect retail investors, European Securities and Markets Authority, 27 March 2018. The source of the 74 to 89 percent figure quoted on this page: national regulators' reviews of retail CFD accounts across EU jurisdictions.
COBS 22.5: Restrictions on the retail marketing, distribution and sale of contracts for differences, Financial Conduct Authority Handbook. The UK rule requiring each firm to publish the percentage of its own retail client accounts that lost money, recalculated every three months.

Frequently asked questions

How do you calculate expected value?

Multiply each outcome by its probability and add the results. For a repeated bet with two outcomes that is EV = (win rate x average win) minus (loss rate x average loss). A 60% win rate with a $400 average win and a $250 average loss gives (0.60 x 400) minus (0.40 x 250) = $240 minus $100 = $140 per trade. Then subtract whatever the trade costs you, because spread, commission, and fees are charged on every repetition whether it wins or loses.

What win rate do I need with a 2:1 risk reward ratio?

If your average win is twice your average loss, you break even at 33.3%. The formula is average loss divided by (average win plus average loss), so 1 / (2 + 1) = one third. Every win rate above 33.3% is profit before costs. Flip it around and the point becomes sharper: if your average win is half your average loss, you need 66.7% just to stand still, which is why a high win rate on its own proves nothing.

Is positive expected value a guarantee?

No. Expected value is the average you converge on over many repetitions, and it says nothing about any single one. A strategy with a genuinely positive edge can lose for weeks, because the standard deviation of one trade is usually several times its expected value. That is why this page shows a range over your chosen number of trades rather than a single total, and why it tells you how many repetitions the edge needs before it stands clear of ordinary bad luck.

How many trades before I know my strategy works?

This page uses a simple, stated rule: the run length at which your expected total is two standard deviations clear of zero, because at that separation a break-even strategy would only match your result about 2% of the time by chance. In symbols, n = 4 x variance / EV squared. A believable retail edge, say 55% at one to one with modest costs, needs around 1,584 trades under that rule. If your inputs say you would know in 30 trades, the inputs are the thing to check.

Why does this calculator ask for the average loss as a positive number?

Because the formula already subtracts it. If you type a minus sign into the average loss box, the loss gets counted twice and the answer comes out far better than the truth, which is the single most common way this arithmetic goes wrong on a spreadsheet. Enter 250 for a $250 average loser, not -250. The calculator handles the sign, and it refuses a negative rather than quietly flattering you.

Do trading costs really change the answer that much?

It depends entirely on the size of your targets, because costs are roughly fixed per trade while profit scales with the target. Take the 60% strategy above and shrink both the win and the loss by a factor of ten, to a $40 average win and a $25 average loss. The gross expected value falls to $14 a trade, and a $15 round turn turns it into a $1 loss. Same edge, same ratio, opposite sign, and the break-even win rate jumps from 38.5% to 61.5%.

What does expected value not tell me?

How much to risk. Expected value is a per-repetition average with no opinion about position size, and betting too large on a real edge is the classic way to be right and broke at the same time, because a bad run can reach zero before the long run arrives. Risk of ruin and Kelly criterion sizing are the tools for that question. This page does not compute them, and no EV figure substitutes for them.

Can I use this for a bet, a game, or a business decision?

Yes, and the second mode is built for exactly that. Enter up to six outcomes with their probabilities and you get the textbook expected value, plus the variance and standard deviation that tell you how bumpy the ride is. It works for dice, insurance, a product launch with three scenarios, or a single number on an American roulette wheel, which pays 35 to 1 on a 1 in 38 chance and therefore returns about 94.7 cents on the dollar.