Flip a coin 10 times. How many heads? Could be 0, could be 10 — but you'd bet on 5. The binomial distribution turns that hunch into exact probabilities.
The binomial distribution describes the number of successes in n independent trials, each with the same success probability p. Coin flips, free throws, defective parts on a line, click-throughs — anything that's 'try n times, count the wins'.
Quality control sampling, A/B testing, polling, genetics, reliability engineering — wherever you count successes out of a fixed number of attempts.
0 balls dropped through 10 rows of pegs. The dashed line is the theoretical normal curve — the histogram converges to it as you drop more.
C(n, k) counts the ways to pick which k trials succeed; pᵏ(1−p)ⁿ⁻ᵏ is the probability of any one such arrangement.
Key facts
- Mean = np — expected number of successes.
- Variance = np(1−p); standard deviation = √(np(1−p)).
- Symmetric when p = ½; skewed otherwise.
- For large n it's well-approximated by a normal distribution (that's the Galton board above).
A basketball player makes 80% of free throws. What's the chance she makes exactly 3 of 4?
10 fair coin flips — probability of exactly 5 heads?
C(10,5)·(½)¹⁰ = 252/1024 ≈ 0.246. The single most likely outcome — but still under 25%.
The trials must be independent and p constant. Drawing cards without replacement breaks this — that's the *hypergeometric* distribution, not binomial.
Add up many binomials (or run a Galton board with more rows) and you get the bell curve. The binomial is the normal distribution's discrete ancestor.
- Counts successes in n independent trials, each with probability p.
- P(X = k) = C(n, k) pᵏ (1−p)ⁿ⁻ᵏ; mean = np.
- Approaches the normal distribution for large n — see the Galton board.