Binomial P(X=k)

Binomial probability P(X = k)

The binomial distribution counts how many successes you get in n independent Bernoulli trials, where each trial succeeds with probability p: P(X = k) = C(n, k) · p^k · (1−p)^(n−k), with C(n, k) = n! / (k!(n−k)!) being the binomial coefficient. The mean works out to μ = n·p and the variance to σ² = n·p·(1−p). Take 10 tosses of a fair coin and ask for exactly 7 heads: C(10, 7) · 0.5⁷ · 0.5³ = 120/1024 ≈ 0.117, about 11.7%. Once n is large enough that np ≥ 5 and n(1−p) ≥ 5, a Normal(np, np(1−p)) gives a good approximation, which follows from the Central Limit Theorem.

Applications

A/B testing leans on it to count conversions, quality control to count defective items in a batch, and Mendelian genetics to predict offspring proportions. You'll also see it behind opinion polling (the margin of error for a proportion), digital communication (bit error rate, or BER), clinical trials measuring response to treatment, and reliability engineering, where it gives the chance of k failures among n components.

FAQ

When does the binomial model not apply? Two cases break it. One is when the trials aren't independent, such as sampling without replacement from a small population, where the hypergeometric is the right model instead. The other is when p shifts from trial to trial.

Binomial or Poisson? Reach for Poisson when n is very large and p very small, yet np stays moderate. That's the rare-events regime, and Poisson tracks the binomial closely there.

What's P(X ≤ k) vs P(X = k)? P(X = k) is the chance of getting exactly k successes. P(X ≤ k) is the cumulative version, the sum of P(X = i) from i = 0 up to k, which answers "at most k successes."