# Binomial and Bernoulli Distributions > The binomial distribution gives the probability of a number of wins in a set of trades. Learn the formula, trading examples and its link to option trees. Source: https://learn.tradelabsai.com/math/binomial-distribution/ Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03 Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice. Cite as: TradeLabs Learn, "Binomial and Bernoulli Distributions", https://learn.tradelabsai.com/math/binomial-distribution/ The binomial distribution answers a common trading question: if each trade wins with a certain probability, how likely is it to get a given number of wins out of a set number of trades? It applies whenever you have a fixed number of independent trials, each with two outcomes and the same probability of success. Traders use it to judge whether a run of results is unusual, to understand how much win rates vary and as the building block of binomial option pricing trees. ## The formula ``` P(k wins in n trials) = C(n, k) × p^k × (1 - p)^(n - k) C(n, k) = n! / (k! × (n - k)!) ``` - **n:** number of trades - **k:** number of wins - **p:** probability of winning each trade ## Mean and standard deviation ``` mean = n × p standard deviation = √(n × p × (1 - p)) ``` ## Worked examples **Example: Wins in 20 trades** A strategy wins 60% of the time. In 20 trades: - Expected wins: 20 × 0.6 = 12. - Standard deviation: √(20 × 0.6 × 0.4) ≈ 2.19 wins. - Probability of exactly 12 wins: about 18%. - Probability of 9 or fewer wins (45% or worse): about 13%. So roughly one time in eight, a genuine 60% strategy will show a 45% or worse win rate over 20 trades. Short term underperformance is normal. See [Losing and Winning Streaks](https://learn.tradelabsai.com/risk/losing-and-winning-streaks/). **Example: Is a track record unusual?** A trader claims a 50% coin flip strategy produced 15 wins in 20 trades. The probability of 15 or more wins in 20 at p = 0.5 is about 2.1%. That is unusual, but if many traders try similar strategies, someone will get such results by chance. See [P-Hacking and Multiple Testing](https://learn.tradelabsai.com/research/p-hacking-and-multiple-testing/). ## Binomial and the win rate The observed win rate k / n has a standard deviation of √(p(1 minus p) / n), the standard error of a proportion. That is why short samples of trades give unreliable win rate estimates. See [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/). ## The normal approximation For large n, the binomial distribution looks like a normal distribution with the same mean and standard deviation. A common rule is that the approximation works when both np and n(1 minus p) are at least about 10. ## Binomial option pricing The binomial option pricing model, developed by Cox, Ross and Rubinstein in 1979, assumes the price moves up or down by fixed factors at each step. After many steps, the number of up moves follows a binomial distribution, and option values are calculated by working back through the tree. As steps increase, the distribution of prices converges to a lognormal shape. See [Binomial and Trinomial Trees](https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/) and [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/). ## Prediction markets and binomial thinking If you make 50 independent prediction market bets at prices near 50 cents, each correct with probability 0.55, the number of correct bets follows a binomial distribution with mean 27.5 and standard deviation about 3.5. Even with a real edge, you could easily get only 24 correct and lose money. Sizing and sample size matter. See [Prediction Market Strategies and Risks](https://learn.tradelabsai.com/prediction-markets/prediction-market-strategies/). ## Limits - **Independence:** real trades are often correlated, especially in the same market or regime. - **Constant probability:** win rates change with conditions. See [Structural Breaks and Regime Changes](https://learn.tradelabsai.com/math/regime-changes/). - **Win or lose only:** the binomial ignores the size of wins and losses, which matter for profitability. See [Expected Value](https://learn.tradelabsai.com/math/expected-value/). ## Frequently asked questions ### What is the binomial distribution? The probability distribution of the number of successes in a fixed number of independent trials, each with the same probability of success. ### How is the binomial distribution used in trading? To calculate the probability of different numbers of winning trades, judge whether results are unusual and build binomial option pricing models. ### What is the standard deviation of wins in n trades? The square root of n × p × (1 minus p), where p is the win probability. Next, learn the distribution of rare events in [Poisson and Exponential Distributions](https://learn.tradelabsai.com/math/poisson-distribution/). ## Continue learning - Next lesson: [Poisson and Exponential Distributions](https://learn.tradelabsai.com/math/poisson-distribution/) - Previous lesson: [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/) - Related: [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/): The Student's t distribution has fatter tails than the normal. Learn how it is used for small sample tests and to model fat tailed returns, with examples. - Related: [Probability for Traders](https://learn.tradelabsai.com/math/probability-for-traders/): Probability is the language of uncertainty in trading. Learn the core rules, independent vs dependent events, odds, and how probability shapes sizing and edge. - Related: [Losing and Winning Streaks](https://learn.tradelabsai.com/risk/losing-and-winning-streaks/): Losing streaks are a normal part of any strategy. See how long streaks get at different win rates, why they happen and how to handle them without breaking rules. - Related: [Binomial and Trinomial Trees](https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/): Binomial and trinomial trees price options by stepping prices up and down through time. Learn how the Cox Ross Rubinstein model works, with a worked example. - Related: [Sampling and Standard Error](https://learn.tradelabsai.com/math/sampling-and-standard-error/): Standard error measures how much an estimate like a win rate or average return varies between samples. Learn the formulas and what they mean for backtests. - Related: [Poisson and Exponential Distributions](https://learn.tradelabsai.com/math/poisson-distribution/): The Poisson distribution models how many rare events occur in a period, like large moves or trade arrivals. Learn the formula, examples and its limits.