Binomial Distribution

Binomial distribution

The binomial distribution tells you how likely you are to get exactly k successes across n independent Bernoulli trials, each one succeeding with probability p: P(X = k) = C(n,k) · p^k · (1−p)^(n−k), where C(n,k) is the binomial coefficient "n choose k". The mean is μ = n·p and the variance is σ² = n·p·(1−p). Say you flip a fair coin 10 times. The chance of landing exactly 7 heads is C(10,7) · 0.5⁷ · 0.5³ ≈ 0.117, or 11.7%. Once n gets large and both np and n(1−p) clear 5, a Normal(np, np(1−p)) approximates it well, which is the Central Limit Theorem at work.

Applications

Think A/B testing on conversion rates, pass-or-fail counts in exams and selection processes, quality control on defects in a batch, Mendelian inheritance proportions in genetics, opinion polls and the confidence intervals built around proportions, reliability engineering when you want the odds of k failures among n components, and clinical trials measuring response to treatment.

FAQ

When does the binomial model not apply? Two cases break it. One is dependent trials, like sampling without replacement from a small population, where you should switch to the hypergeometric. The other is when p drifts from trial to trial.

Binomial or Poisson? Reach for Poisson when n is very large and p is very small while np stays moderate, the classic rare-event setup. In that regime Poisson tracks the binomial closely.

What's P(X ≤ k)? That is the cumulative distribution, the sum of P(X = i) for i running from 0 to k. You use it to answer "probability of at most k successes".