EV / Variance Calculator

Why single-outcome calculators fail on real bets

Most betting calculators handle one scenario: you either win or you lose. Real wagers refuse to be that tidy. A roulette player covering corners and splits holds several simultaneous outcomes. A sports bettor hedging a futures ticket with a live underdog has created a new paytable from scratch. Video poker and slots publish dozens of prize tiers.

Knowing only your win probability hides half the picture. Assign an exact probability to every payout in the table and you get the true average return of the whole structure — not a rough guess built on one row.

The math: Expected Value, Variance, and Standard Deviation

Four formulas sit behind every result this calculator produces. Here they are, in the order the tool applies them:

1. Expected Value (EV)

EV is the probability-weighted average across every outcome — what you expect to win or lose per bet over a long series:

EV = (p1 * x1) + (p2 * x2) + ... + (pn * xn)

Here p is a decimal probability between 0 and 1, and x is the net payout for that outcome.

2. Variance (Var)

Variance measures how widely results scatter around that average. Low variance keeps returns near the EV; high variance means the balance lurches:

Variance = (p1 * (x1 - EV)^2) + (p2 * (x2 - EV)^2) + ... + (pn * (xn - EV)^2)

An equivalent shortcut form, which the calculator uses internally:

Variance = Σ (p_i * x_i^2) - EV^2

3. Standard Deviation (σ)

Variance arrives in “squared dollars,” which nobody budgets with. Taking its square root converts the figure back into ordinary currency:

Standard Deviation (σ) = √Variance

4. Coefficient of Variation (CoV)

CoV expresses volatility as a ratio against expected return, which makes bets of different sizes comparable:

CoV = σ / |EV|

A high CoV flags a wager where risk dwarfs the expected profit margin — the classic shape of a lottery ticket.

Data Sandwich: Scaling volatility over N rounds

Suppose you’ve found a positive-EV roulette hedge and run it through the calculator: +$2 expected per spin, $15 standard deviation. Good start.

Over 100 spins the expectation reaches $200. Risk behaves differently — it grows with the square root of bet count, not linearly:

σ_N = σ * √N

After 100 spins your standard deviation is $15 * √100 = $150.

Under normal distribution rules, roughly 68% of 100-spin sessions land within one standard deviation of the mean — here, between $50 and $350. That leaves plenty of sessions finishing below zero despite a genuine edge. Shrinking that downside requires volume: thousands of wagers before variance stops dictating the outcome and expectation takes over.

Frequently asked questions