Variance in Betting: Expected Value, Paths and Drawdown

Binary calculation

Let one unit win one unit with probability p = 0.55 and lose one unit with probability 0.45. The illustrative outcome X is +1 or -1.

Expected value:

E[X] = 0.55 * 1 + 0.45 * (-1) = 0.10 units

Second moment:

E[X^2] = 0.55 * 1^2 + 0.45 * (-1)^2 = 1

Variance:

Var(X) = E[X^2] - E[X]^2 = 1 - 0.10^2 = 0.99

Standard deviation = sqrt(0.99) = approximately 0.994987 units

OpenStax provides the expected-value and standard-deviation definitions. The example assumes the stated p is correct; it does not establish a real betting edge.

Sequence probability needs assumptions

In this illustrative model, under independent trials with the same 45% loss probability, five consecutive losses have probability:

0.45^5 = 0.0184528, or about 1.845%

That is the probability for one specified five-bet block under the model. It is not the chance of seeing some five-loss run anywhere in a long history, and real bets may not be independent or identically distributed. OpenStax's independence guidance states the multiplication condition used here.

Variance, uncertainty, and drawdown differ

The distinctions below use the expected-value definitions above and a formal ruin model in Ferguson (1963):

• Outcome variance describes spread under a stated distribution.
• Estimation uncertainty concerns whether probabilities and payoffs were estimated correctly.
• Drawdown is a path statistic from a previous bankroll peak.
• Ruin requires a defined threshold, staking rule, horizon, and stochastic model.

Ferguson's formal ruin analysis shows why bankroll, costs, minimum bets, and stopping conditions belong in the model rather than in a universal slogan.

Report the path

For each reviewed series, publish net result, total stake, yield, maximum drawdown, longest losing run, largest open exposure, and the odds and stake distribution. Add uncertainty around the mean result. NIST's confidence-limit guidance explains why higher sample variability widens uncertainty for a mean.

Do not select only the worst drawdown from one method and the average return from another. Use the same period, decisions, settlement, and bankroll definition.

What variance cannot excuse

Do not label every loss as variance. Recheck probability calibration, accepted prices, omitted selections, execution, and rule changes. A negative later-period result may be compatible with a positive model expectation, but it is still evidence to quantify, not dismiss.

Publish a simulation release record

If Monte Carlo is used, publish the payoff distribution, probability source, stake rule, number of paths, horizon, dependence model, random seed, minimum stake, and ruin or drawdown thresholds. Report multiple percentiles and the proportion of paths that cross each threshold. Do not discard paths that reach a stake limit or zero balance.

Validate the simulator against boundary cases with known answers, including zero variance, fair even-money trials, and a no-bet policy. Repeat with adverse probabilities and clustered losses. Simulation samples the model supplied; it cannot establish that the model probabilities, dependence, or future market conditions are correct.

Next step

Use Betting Roi Calculator for the next part of this topic.

Continue learning

• Next guide: Level Stakes vs Variable Stakes
• Related guide: Bet Sizing Under Probability Uncertainty

Assumptions and limitations

The examples use independent binary unit outcomes and known probabilities. Football prices, stakes, pushes, commission, and dependencies vary. Historical variance does not guarantee future variance, and no sample size removes model risk.