Expected Value Calculator
Calculate the mathematical expected value (EV), return per wager, and expected net profit or loss across single or multi-outcome betting scenarios.
Expected value represents long-run statistical expectation across comparable trials—not a prediction of your next outcome.
Last reviewed: October 2026
Methodology: Discrete Random Variable Expected Value ($EV = \sum P_i \times x_i$)
Under the probabilities and payout values specified, the mathematical expectation is exactly neutral ($0.00 per wager).
What Is Expected Value?
In probability theory and casino mathematics, Expected Value (EV) measures the long-run average result of a random variable across a large number of repeated trials.
When applied to gambling or betting, expected value calculates the average net amount a player expects to win or lose per wager under the specified probabilities and net payouts.
Expected Value Formula
EV: Expected Value per wager
Σ: Sum of all possible mutually exclusive outcomes
Pᵢ: Probability of outcome i occurring ($0 \le P_i \le 1$)
xᵢ: Net monetary gain or loss for outcome i
What Does Expected Value Mean?
EV > 0
The mathematical expectation is a positive net gain per wager under the stated model. Over a massive sample size, the player holds a theoretical advantage over the house or market.
EV < 0
The mathematical expectation is a negative net result per wager. Most standard casino wagers carry negative expected value due to the built-in house advantage.
EV = 0
The mathematical expectation is break-even. Neither the player nor the house holds a statistical advantage over time under theoretical probabilities.
How to Calculate Expected Value
Follow these six mathematical steps to calculate the expected value of any gamble, bet, or decision with discrete outcomes:
- 1. Identify All Outcomes Define every mutually exclusive outcome that can occur during the trial.
- 2. Assign Probabilities Determine the exact probability ($P_i$) for each outcome. The total probability sum must equal 1.0 (or 100%).
- 3. Express Net Results Convert payouts into net gains or losses ($x_i$). Subtract the original wager from gross payouts so the stake is not double-counted.
- 4. Multiply Probability × Net Result For every outcome, calculate $P_i \times x_i$.
- 5. Sum the Weighted Results Add all individual weighted products together to arrive at the overall EV.
- 6. Interpret EV per Unit Wager Divide EV by the initial stake to express EV as a percentage of the wagered amount.
Worked 3-Outcome Example
Suppose a game involves a $50 base wager with three potential results:
| Outcome | Probability ($P_i$) | Net Result ($x_i$) | Weighted Value ($P_i \times x_i$) |
|---|---|---|---|
| Jackpot Win | 20% (0.20) | +$100.00 | 0.20 × $100 = +$20.00 |
| Small Win | 30% (0.30) | +$20.00 | 0.30 × $20 = +$6.00 |
| Loss | 50% (0.50) | -$50.00 | 0.50 × (-$50) = -$25.00 |
Expected Value of a Bet
In sports betting and fixed-odds gaming, expected value evaluates whether the payout odds offered by a bookmaker or operator reflect true mathematical probability.
EV in Dollars
The net monetary expected outcome ($) for a specific stake size. Scaling the stake increases dollar EV linearly without changing the EV percentage.
EV Per Unit Stake
The expected return expressed per 1.0 unit wagered (e.g., +0.02 units expected profit per 1 unit wagered).
EV Percentage
The mathematical advantage or disadvantage relative to total money wagered ($EV\% = \frac{\text{EV}}{\text{Stake}} \times 100$).
Expected Value in Casino Games
Casino games are designed with underlying mathematical rules, paytables, and wheel or deck configurations that dictate exact probability distributions. Below is a breakdown of how EV functions across major game formats:
Roulette Expected Value
Roulette expected value depends on whether the wheel features a single zero (European) or double zero (American), as well as specific rule variations.
- European Single-Zero Straight Up: EV = -2.70% (-$0.27 per $10)
- American Double-Zero Straight Up: EV = -5.26% (-$0.53 per $10)
Blackjack Expected Value
Blackjack does not have a single fixed EV. Expected value varies dynamically depending on table rules (3:2 vs 6:5 payouts, deck count, dealer soft 17 rules) and player decisions.
- 3:2 Blackjack with Perfect Basic Strategy: EV ~ -0.50%
- 6:5 Blackjack Payout Penalty: Lowers EV by ~1.39%
Baccarat Expected Value
Baccarat wagers carry distinct probabilities and commission adjustments. The Banker wager yields the highest mathematical expectation despite the standard 5% commission.
- Banker Bet (5% commission): EV = -1.06%
- Player Bet: EV = -1.24%
- Tie Bet (8:1 payout): EV = -14.36%
Slot & Video Poker Expected Value
Slot EV is derived directly from the reel strip geometry, symbol combinations, and bonus configurations programmed by developers. Video Poker EV is dictated by paytable variations (e.g., 9/6 full-pay Jacks or Better yields ~99.54% expected return under optimal strategy).
Expected Value vs RTP
Return to Player (RTP) is expressed as a percentage of total turnover returned across millions of spins or hands. Expected Value (EV) measures the expected net dollar or unit result per wager.
For instance, a game with a 96% RTP carries an EV percentage of -4.00% per dollar wagered.
Expected Value vs House Edge
House Edge represents the mathematical advantage the casino holds over the player, expressed as a positive percentage of the initial wager.
If European Roulette has a house edge of 2.70%, the player's expected value per wager is -2.70% (or -$0.27 per $10 stake).
Expected Value vs Variance & Volatility
Expected Value and Variance address fundamentally different characteristics of a probability distribution:
Answers: What is the long-term mathematical center or average outcome of the random process?
Answers: How widely can individual session outcomes spread around that mathematical average?
Two games can share the exact same expected value (e.g., -2.70% EV), yet exhibit radically different short-term swings. A low-volatility bet returns small, frequent outcomes, whereas a high-volatility bet concentrates payouts into rare, large spikes.
Can a Positive EV Bet Lose?
Yes, absolutely. Having positive expected value (+EV) does not guarantee that a single bet will win. For instance, a bet with a 60% win probability paying 1:1 carries a positive EV (+0.20 per dollar), but will still lose 40% of the time on individual trials.
Can a Negative EV Bet Win?
Yes, frequently in the short run. A single-zero roulette spin has a negative EV (-2.70%), but hitting a straight-up number produces an immediate +35:1 net profit. Random variance allows short-term gains despite underlying negative expectation.
Why Actual Results Differ From Expected Value
Observed gambling session results almost always deviate from theoretical expected value due to the following statistical forces:
- Finite Sample Size: Convergence toward theoretical expected value requires a law of large numbers scale (tens of thousands or millions of trials). A session of 100 hands is statistically microscopic.
- Standard Deviation & Variance: Short-term random noise naturally dominates small sample sizes, causing actual returns to fluctuate in wide confidence bands around EV.
- Independence of Events: Past outcomes do not influence future random trials. A negative-EV game never becomes "due" to adjust toward its EV during a short window.
How to Calculate EV Percentage
To normalize EV across varying stake sizes, calculate EV percentage using total initial wager as the denominator:
Example: If a $20 wager carries a monetary expected value of -$0.54, the EV percentage is $\left( \frac{-0.54}{20} \right) \times 100 = -2.70\%$.
Frequently Asked Questions
What is an Expected Value Calculator?
What does EV mean in gambling?
How do you calculate expected value of a bet?
What does positive expected value (+EV) mean?
What does negative expected value (-EV) mean?
Can a positive EV bet lose?
Can a negative EV bet win?
What is the difference between EV and RTP?
What is the difference between EV and house edge?
Does expected value predict the next bet?
How does stake size affect EV?
Does variance affect expected value?
Related Casino Mathematics Tools
Explore complementary probability and statistical analysis calculators.
RTP Calculator
Calculate theoretical Return to Player percentages.
House Edge Calculator
Determine exact casino house advantages across rules.
Casino Loss Calculator
Estimate theoretical loss based on volume and edge.
Probability Calculator
Compute discrete outcome probabilities.
Casino Volatility Research
Understand standard deviation and short-term variance.
How Adzvelo Calculates Expected Value
Methodology, mathematical limits, and transparent assumptions.
Adzvelo's Expected Value Calculator applies discrete probability summation ($EV = \sum P_i \times x_i$). Outcomes are evaluated as net returns relative to the initial stake amount to prevent stake double-counting.
Disclaimer: Adzvelo's Expected Value Calculator is a mathematical modeling tool. It does not independently test casino games, certify game fairness, or predict future gambling outcomes.
Game-specific EV examples cited on this page assume standard game rules, official paytables, or recognized probability distributions. Real-world gaming results depend on operator configurations and adherence to optimal strategy where applicable.
Sources & Authoritative References
Responsible Gambling Statement
Expected Value is a mathematical expectation calculated over infinite trials and does not guarantee financial profit or protect against monetary loss in short-term sessions. Gambling outcomes remain fundamentally uncertain. Never treat expected value calculations as financial advice or predictions of future results.