Most gamblers judge a wager by the size of the jackpot. Professional bettors ignore the noise and look at two numbers: expected value and variance (read our theoretical guide to Expected Value Explained). This Expected Value (EV) and Variance Calculator lets you enter multi-outcome scenarios to find your exact return and standard deviation.
| Probability (0–1) | Payout (units) | |
|---|---|---|
Simple betting calculators assume a binary world: you win or you lose. But real betting is rarely that neat. In roulette, you might hedge with corner bets and split bets. In sports betting, you might hedge a futures ticket with a live underdog. In casino games like video poker or slots, you have a massive paytable of varying prize sizes.
If you only calculate your win probability, you miss the actual value of the bet. By mapping every possible payout with its exact probability, you see the true average return of your strategy.
Calculating the average return and volatility of a complex bet requires a few standard steps. Here is how the calculator processes your multi-outcome inputs:
Expected Value is the probability-weighted average of all possible outcomes. It is the average amount you expect to win (or lose) per bet over the long run:
EV = (p1 * x1) + (p2 * x2) + ... + (pn * xn)
Where p is the probability of an outcome (expressed as a decimal between 0 and 1) and x is the net payout for that outcome.
Variance measures how far the actual outcomes are spread out from the expected average. A low variance means your returns will be stable and close to the EV. A high variance means your balance will swing violently:
Variance = (p1 * (x1 - EV)^2) + (p2 * (x2 - EV)^2) + ... + (pn * (xn - EV)^2)
Or calculated efficiently as:
Variance = Σ (p_i * x_i^2) - EV^2
Because variance is in “squared dollars,” we take the square root of the variance to get the Standard Deviation. This returns the risk metric back into a regular currency value:
Standard Deviation (σ) = √Variance
The Coefficient of Variation measures relative volatility. It is the ratio of standard deviation to the expected return:
CoV = σ / |EV|
A high CoV indicates that the risk of the wager is extremely high compared to the expected profit margin.
Let’s say you identify a positive EV roulette hedging system. After entering your bets, the calculator tells you that your expected profit is +$2 per spin, with a standard deviation of $15.
If you play 100 spins, your expected return is $200. However, your risk does not scale linearly; it scales with the square root of the number of bets:
σ_N = σ * √N
For 100 spins, your standard deviation scales to $15 * √100 = $150.
Using standard normal distribution rules, about 68% of your 100-spin sessions will end between one standard deviation below and above your expected profit (between $50 and $350). There is still a significant chance you will finish in the negatives, despite holding an edge. To smooth out the volatility and guarantee a profit, you must scale your sample size to thousands of wagers.
A negative expected value means that the game is mathematically stacked against you. For every dollar wagered, you will lose a percentage on average over time. Almost all standard casino games are -EV for the player due to the house edge.
A positive EV wagers can still ruin you. If you have a highly profitable 10% edge but a massive standard deviation, a brief sequence of bad luck can wipe out your entire bankroll before the law of large numbers saves you. Variance dictates your bet sizing.
Make sure you enter net winnings in the payout column, not the total return. For example, if you place a $10 bet at 2:1 odds, your net payout on a win is +$20 (winnings), and your net payout on a loss is -$10 (the lost stake).