The Bernoulli Distribution

A Bernoulli random variable takes only two values, conventionally $1$ ("success") and $0$ ("failure"), with success probability $p$ and failure probability $1-p$. It's the mathematical model for a single yes/no event: one coin flip, whether one specific trade is a winner, whether one specific option expires in-the-money.

Its mean is simply $E[X] = p$ — the long-run average of a 0/1 variable is just the probability of getting a 1. Its variance is $\mathrm{Var}(X) = p(1-p)$, which is largest when $p = 0.5$ (maximum uncertainty about the outcome) and shrinks toward zero as $p$ approaches 0 or 1 (the outcome becomes near-certain either way).

Worked example: a strategy wins a given trade with probability $p = 0.6$. Its per-trade outcome has mean $0.6$ and variance $0.6 \times 0.4 = 0.24$, so a standard deviation of about $0.49$ — nearly as large as the mean itself, which is typical for Bernoulli outcomes near the middle of the range. Summing $n$ independent Bernoulli trials with the same $p$ gives a binomial random variable, the count of successes out of $n$ tries; this is exactly how a win-rate estimate over a backtest is built, one Bernoulli outcome per trade.