Table of Contents
What Is the Binomial Distribution?
The binomial distribution models the number of successes in n independent Bernoulli trials, each with the same probability p of success. It answers questions like "What is the probability of getting exactly 7 heads in 10 coin flips?" or "What is the probability that 3 out of 5 patients respond to treatment?"
It requires: fixed number of trials, two outcomes (success/failure), constant probability, and independence between trials.
Formula
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
Mean = np | Variance = np(1-p)
Example: n=10, p=0.5
| k | P(X=k) | P(X≤k) |
|---|---|---|
| 0 | 0.0010 | 0.0010 |
| 3 | 0.1172 | 0.1719 |
| 5 | 0.2461 | 0.6230 |
| 7 | 0.1172 | 0.9453 |
| 10 | 0.0010 | 1.0000 |
FAQ
When does binomial approximate normal?
When np ≥ 5 and n(1-p) ≥ 5, the normal approximation N(np, np(1-p)) is reasonable.
Binomial vs Poisson?
Use Poisson when n is large and p is small (rare events). Poisson approximates binomial with λ = np.