Variance Calculator

Leave Win Probability at 48%, Decimal Odds at 2.0, Bet Size at $10, and Trials at 1000. Calculate. Standard Deviation is $315.97, Variance is $99,840.00, and Expected Profit is −$400.00. Those three numbers describe a thousand independent even-money bets that win 48% of the time. Each win nets +$10 (odds 2.0 means even money: you receive $20 including stake). Each loss is −$10. The per-bet mean is 0.48×10 + 0.52×(−10) = −0.40, so a thousand trials expect −$400. The per-bet second moment around that mean is 99.84 dollars-squared; multiply by 1000, take the square root, and you have a session SD of $315.97.

That SD is the practical output. A roughly two-thirds band around the mean (if you trust a normal approximation after 1000 Bernoulli trials) runs from −400 − 316 to −400 + 316, i.e. about −$716 to −$84. You can finish a thousand 48% even-money bets in the black; it is a right-tail event, not the centre. Changing Trials to 4000 multiplies variance by four and SD by two ($631.95) while expected profit goes to −$1,600. The mean leaks four times faster than the SD grows, which is the law of large numbers in one table.

Odds are decimal. 2.0 is even money. 3.0 is 2-to-1: a $10 winning bet nets $20, not $10. At 48% and 3.0 with $10 and 1000 trials the mean flips positive (0.48×20 + 0.52×(−10) = 4.4 per bet, +$4,400 per thousand) and the SD inflates because the win and loss atoms are further apart. If you type American −110 as 1.91, do the conversion first; this field is not American odds.

Win Probability is a percent, 0 to 100. A coin is 50. A 1.91-priced football side that you believe is 52% goes in as 52, not 48. The 48% default is a slightly-off coin, the typical shape of a small house edge on an even-money casino bet (pass-line is 244/495 ≈ 49.3% after pushes are removed, but 48% is a clean classroom number).

Bet Size scales both mean and SD linearly. Trials scale mean linearly and SD with the square root. If you want the per-bet SD instead of the session SD, set Trials to 1: you will see $9.99 of standard deviation on a $10 even-money 48% bet, which is almost the full stake — as De Moivre would have told a 1718 gambler.

Variance is the second moment of your P&L, the quantity that turns a −$0.40 expected leak into a session that can still print black or still go three times worse than the mean. Gamblers who quote only RTP or house edge are quoting the first moment. Bankrolls die on the second. This calculator takes a two-outcome bet — win with probability p at decimal odds o, lose the stake otherwise — repeats it n times, and reports the exact mean, variance, and standard deviation of the sum under independence.

The intellectual history is Abraham de Moivre's Doctrine of Chances (1718) and the Bernoulli theorem that the sample mean concentrates, with fluctuations of order 1/√n. A casino is a factory that sells you n of these trials. The factory's edge shows up in the mean; the factory's survival shows up in the fact that n is large enough for their √n noise to be small relative to n×edge, while your n is not. Over 1000 even-money 48% bets the −$400 mean and $316 SD are the same order, so luck still dominates. Over 100,000 bets the mean is −$40,000 and the SD is about $3,160, and luck is a rounding error.

Two-outcome variance is the building block underneath more exotic paytables. A slot is a mixture of many atoms; its variance is Σ p_i (x_i − μ)², which this tool will not compute. If you collapse a slot to "hit / miss" you have thrown away the royal-flush-sized atom that dominates the second moment. Use this calculator for coin-flip-shaped bets: pass-line, banker baccarat (approximately), a sports side, a handicap. Do not use it for keno, slots, or a parlay whose payoff is a product of legs — those need a different generating function.

Independence is the hidden assumption. A finite-shoe blackjack count, a Martingale that resizes the next stake, or a parlay built from correlated legs all break n × Var(one bet). The output will still multiply by Trials; it will be wrong. Sequential bets with a stop-loss are not n i.i.d. copies of the original bet either. This is a fixed-stake, fixed-odds, i.i.d. engine.

Limits: no risk of ruin, no Kelly path, no higher moments (skew of a 48% even-money bet is mild; skew of a 1-in-1000 longshot is not, and the normal band quoted in How to Use becomes a lie). Read Standard Deviation as the L2 width of the P&L, not as "maximum drawdown." Drawdown is a path statistic. Variance is a single number.

Variance is the second moment of profit and loss, the quantity that turns a small expected leak into a session that can still print black or still go several times worse than the mean. Gamblers who quote only return to player or house edge are quoting the first moment. Bankrolls die on the second. Abraham de Moivre’s Doctrine of Chances and the Bernoulli theorem already said that the sample mean concentrates, with fluctuations of order one over square root of n. A casino sells you n of these trials. Their edge shows up in the mean; their survival shows up because their n is large enough for square-root noise to be small relative to n times edge, while yours is not.

The two-outcome model is elementary. Win probability p is the typed percent divided by 100. Decimal odds o convert a stake B into net win W = B(o − 1) and net loss L = −B. The per-bet mean is mu = p W + (1 − p) L. The per-bet variance is p times (W − mu) squared plus (1 − p) times (L − mu) squared. Independence across trials multiplies variance by n. Session standard deviation is the square root of that product. Session expected profit is n times mu. That is the entire worksheet: mean, variance, and standard deviation of a sum of independent signed bets.

The defaults are a 48 percent win rate, decimal odds of 2.0, a bet of 10, and 1,000 trials. Odds of 2.0 are even money: a win nets +10 and a loss is −10. Then mu = 0.48 × 10 + 0.52 × (−10) = −0.40. The second moment is 0.48 × (10 − (−0.40)) squared plus 0.52 × (−10 − (−0.40)) squared, which is 0.48 × 10.4 squared plus 0.52 × 9.6 squared, which is 0.48 × 108.16 plus 0.52 × 92.16 = 51.9168 + 47.9232 = 99.84. Session variance is 1,000 × 99.84 = 99,840. Session standard deviation is the square root of 99,840, about 315.97. Session expected profit is 1,000 × (−0.40) = −400. Those three figures are the published outputs: 315.97 of standard deviation, 99,840.00 of variance, and −400.00 of expected profit.

An even-money rewrite is worth keeping. When W = B and L = −B one has mu = (2p − 1)B and per-bet variance = 4 p (1 − p) B squared. At p = 0.48 that is 4 × 0.48 × 0.52 × 100 = 99.84, matching the expansion. Four thousand trials multiply variance by four and standard deviation by two, to about 631.95, while expected profit scales to −1,600. The mean grows like n and the noise like square root of n, so the factory eventually wins by law of large numbers. At n = 1,000 the −400 mean and the 316 standard deviation are the same order: a two-thirds band under a normal approximation runs from about −716 to −84. Finishing in the black is a right-tail event, not the centre.

Assumptions are independence, a stationary p, a full loss of stake, decimal odds that already net the payout, and no pushes. Correlated games (same-shoe blackjack, parlay legs) inflate true variance above this formula. A 40 percent win rate at the same even money would be a different mean, mu = −0.20 per dollar, but the published default is 48 percent, a typical recreational side. The normal approximation is a convenience after a thousand Bernoulli trials; the exact distribution is a shifted binomial. The worksheet reports the exact mean and variance of that sum, then the square root.