Expected value in gambling (worked examples)
Expected value is the average win or loss per bet if the same bet were repeated endlessly, calculated by adding each outcome multiplied by its probability. The formula and six worked examples follow.
The expected value formula
Expected value, or EV, is the average result of a gamble if the same bet could be repeated indefinitely under the same rules. The formula is: EV = (probability of outcome A × result of outcome A) + (probability of outcome B × result of outcome B) + all other possible outcomes.
In words: multiply each possible result by its chance, then add the answers. That is all the formula is doing, even in games that look more complicated.
An expected value calculation can be shown in money or as a percentage of the stake. In money, EV tells you the average dollar result per bet. As a percentage, EV percentage = EV ÷ stake.
The result must use net profit or net loss, not the total amount returned to the player. Negative EV means the bet loses on average before variance.
Positive EV means the price, rules or added value are favourable on the inputs used. This is the core idea behind gambling math: odds, house edge and probability, and it does not predict the next spin, hand or match.
Probability is the part most people get wrong. In casino games, the probabilities are usually fixed by the rules, the wheel, the pay table or the random number generator. In sports, the probability is an estimate, so the quality of the estimate controls the quality of the EV answer.
If the chance is too optimistic, the expected value can look positive while the real bet is poor. EV is only as reliable as the probability and payout figures that feed it.
Expected value also differs from a bankroll plan. A bet can have a known negative EV and still win now. A bet can have positive EV and still lose several times in a row.
EV prices the decision, while variance describes how far the real path can move away from that average. Use the formula to understand the cost or value of a bet, not to claim certainty.
Worked example: roulette
For a European roulette straight-up bet, the expected value is (1/37 × 35) - (36/37 × 1) = -1/37, or -2.70% of the stake. The win side has a chance of 1/37 and a profit of 35 units. The losing side has a chance of 36/37 and loses 1 unit. The minus sign is the house edge expressed from the player’s side.
On a $10 straight-up bet, the same calculation gives an EV of -$0.27. Put differently, the bet has an RTP equivalent of 97.30%. The result does not say any particular spin should lose $0.27, since a straight-up bet either hits or misses. It says the average cost of that $10 decision is -$0.27 if the same bet is repeated under the same wheel rules.
Every European roulette bet works out the same in expectation because the payouts are set against the same wheel. A red or black bet feels smoother than a straight-up number, but the average price is still built into the table.
The difference is volatility, not EV. A high-payout roulette bet has fewer winning outcomes and larger wins. A low-payout roulette bet has more frequent winning outcomes and smaller wins. For the full bet table, see roulette odds and payouts.
Worked example: a blackjack decision
Expected value can also compare choices inside a game. Take a blackjack hand of hard 16 against a dealer 10. Per $10 staked, standing has EV of -$5.40. Hitting also has EV of -$5.40 in this exact example. Surrendering has EV of -$5.00. These figures are decision values for that hand and dealer card, not the overall edge of blackjack.
Those figures do not mean the hand is good. They mean each option is losing in expectation, while surrender loses less than the alternatives where surrender is offered. The best EV decision can still be a negative decision if every available option is bad. That is common in blackjack because the player often faces a least-bad choice rather than a profitable one.
This is why blackjack strategy is a table of conditional decisions. Hard totals, soft totals, pairs, dealer upcards and rule variations all change the calculation. A result for hard 16 against a dealer 10 should not be copied to other hands.
A close EV comparison can also be affected by whether surrender is available at all. For broader decision rules, use a blackjack basic strategy chart.
Worked example: slots and progressive jackpots
A slot’s RTP converts directly to EV. A $1 spin on a slot with 96% RTP has EV of -$0.04, and $1,000 total wagered at 96% RTP has an expected loss of $40. In percentage form, the house edge is the gap between the stake and the return percentage. The link between RTP and house edge is covered in return to player.
That basic RTP example treats the pay table as fixed. Progressive jackpots add a moving component because part of the value depends on the displayed jackpot and the chance of hitting it.
In this example, the slot has 88% base RTP, $1 spins and a jackpot hit chance of 1 in 10,000,000. The break-even jackpot is $1,200,000. At that point, the jackpot contribution offsets the shortfall in the base game.
If the same jackpot shows $800,000, EV is -$0.040 per $1 spin. If it shows $1,500,000, EV is $0.030 per $1 spin. That makes the second jackpot level positive in the arithmetic, but the jackpot remains rare by the rules of the example.
The calculation does not say the jackpot is close to landing. It only prices the average value of a spin at the displayed jackpot amount.
This is the part of slots EV that is easy to misread. A positive EV jackpot situation can still produce long losing stretches because most of the value sits in a rare event.
A player who never hits the jackpot may see results that look nothing like the average. A player who hits it sees an extreme result. The EV number combines those paths into a single average, but the actual experience can be far from smooth.
Worked example: a sports bet
Sports EV starts with the market price and your own probability estimate. Decimal odds of 2.50 imply a probability of 40%. If your estimate is 45% and the stake is $10, the EV is $1.25. That is the positive case in this example because the estimated chance is higher than the chance implied by the odds.
If your estimate is 38% at the same 2.50 price and $10 stake, the EV is -$0.50. The price has not changed, but the probability input has. That is why sports expected value is less mechanical than roulette EV. In roulette, the wheel fixes the chance. In sports, the hard part is estimating the probability without bias.
A useful way to read the result is to ask whether the edge survives a more cautious estimate. A good sports bet has positive EV after using a realistic probability, not after forcing the numbers to support a bet.
A calculator can do the arithmetic quickly, but it cannot prove that the probability estimate is correct. Models, injury information, market movement and judgement can all be wrong. The EV answer should be treated as a pricing tool, not as proof that the bet will win.
Worked example: a casino bonus
Bonus EV uses the same arithmetic, but the result depends on wagering rules. With a $100 bonus at 35x bonus wagering on games with a 4% house edge, the turnover required is $3,500. The expected loss while wagering is $140, so the bonus EV is -$40. In plain terms, the mathematical cost of the wagering is larger than the bonus value in this simplified case.
This is a simplified example. It treats the wagering path as if all required turnover is completed, so it ignores the risk of losing the balance before completion. It also assumes the game contribution and allowed stakes do not block completion.
Real terms can change the usable value of a bonus because some games may contribute less, some bet sizes may be restricted and withdrawals may be conditional. For the mechanics behind these clauses, read wagering requirements explained.
Bonus EV is useful because it turns promotional wording into a costed decision. The headline amount is not the value by itself. The value depends on how much must be wagered, which games count, the house edge of those games and whether the player can complete the requirements without breaking the terms. If any of those inputs changes, the expected value changes with it.
Same EV, different results
Short-run variance explains why players can see very different outcomes while the expected value remains negative. The model below uses a slot with 96% RTP and 27.36% hit frequency, paying 0.5x to 100x, for 100 simulated players making 1,000 spins of $1 each.
The expected result per player is -$40, which comes from the 96% RTP over $1,000 of total staking. The individual results do not line up neatly with that average.
Measure | Result |
|---|---|
Expected result per player | -$40 |
Average result | -$50 |
Median result | -$39 |
Best result | $518 |
Worst result | -$385 |
Players ahead after 1,000 spins | 34 |
EV describes the group average, not the path of an individual player. A player can finish ahead during a negative-EV session, and another can lose more than the expectation. That does not change the underlying price of the game. It explains why short samples can feel convincing while still being mathematically weak.
The best result in the table is positive even though the model game is negative EV. The worst result is far below the expected loss. The median is close to the expected result, while the average is lower in this sample.
That mix is normal in gambling games with uneven payouts. A small number of better outcomes can sit beside many ordinary losses, and the average only becomes meaningful across repeated decisions.
This is also why personal memory is a poor EV calculator. Big wins stand out. Near misses feel meaningful. Losing runs feel due to reverse. None of those impressions changes the expected value. If the rules and payouts stay the same, the EV stays the same. The next result is still random within the game’s rules.
Positive EV situations
Positive EV situations can exist, but they are narrow. Card counting is the classic table-game example because the remaining deck composition can change the player’s expectation. Bonus hunting and promotions can also become favourable if the value offered is greater than the expected cost of clearing it.
Progressive jackpots can be positive in theory when the jackpot amount is high enough relative to the base game and hit chance, as shown in the slot example above.
The limits matter. Casinos can require verification, restrict accounts, exclude games from promotions, set maximum stakes or enforce bonus abuse rules. A calculation that looks favourable before reading the terms can fail once those restrictions are applied.
Positive EV is also not the same as low risk. A favourable but highly volatile opportunity can still lose in the short term, especially where much of the value depends on a rare outcome.
Sports betting has a similar problem. A calculator can compare a price with a probability estimate and show the expected dollar value of a stake. It cannot make the estimate true. If the estimated chance is inflated, the output is misleading. Positive EV is an edge on paper, not a guaranteed result.
Expected value calculator
Enter the chance of winning, the decimal odds and the stake to see the expected value of a single bet.
Keep the calculated average separate from your gambling budget, and set deposit limits, loss limits and reality checks before staking more.
What this means for your stakes
EV tells you the price of a decision. It does not tell you that a session is safe, due to turn, or likely to match the average. If the EV is negative, the cleanest reading is that the entertainment has a mathematical cost. That cost may be small per bet, but repeated staking turns small percentages into real money.
Use expected value examples to compare decisions, not to chase outcomes. In roulette, EV explains why different bet types can have the same average cost with different volatility. In blackjack, it helps rank available decisions. In slots, it connects RTP, jackpots and the chance of uneven results. In sports and bonuses, it shows how sensitive the answer is to assumptions and rules.
No formula removes the need for limits. If a bet is negative EV, staking more increases the expected cost. If a bet appears positive EV, variance can still create losses that exceed what feels comfortable. If gambling starts to feel hard to stop, use gambling helplines and support services by country rather than another calculation.
Frequently asked questions
Can expected value be positive in a casino?
Yes, but only in limited situations such as favourable promotions, rare jackpot conditions or advantage play. The rules and restrictions must create more value than the expected cost. Most ordinary casino bets remain negative EV.
What is +EV betting?
+EV betting means a bet has positive expected value using your estimated probability and the offered price. In sports, that means your fair probability is higher than the probability implied by the odds. It still loses sometimes because EV is an average, not a guarantee.
How do I calculate the EV of a bonus?
Convert wagering terms into turnover, multiply that turnover by the house edge of the games you can use, then subtract expected loss from the bonus value. Also check contribution rules, maximum stake rules and withdrawal conditions. The simplified EV can look better than the real usable value if restrictions prevent completion.
Does expected value guarantee results?
No. EV describes the average over repeated bets with the same rules. Variance can put an individual session above or below that average, sometimes by a wide margin.