Binomial and Bernoulli Distributions

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#

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.

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 and 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.

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.
• Win or lose only: the binomial ignores the size of wins and losses, which matter for profitability. See 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.