Binomial Distribution Formula
Reference for the binomial distribution formula.
Calculate the probability of exactly k successes in n independent trials with worked examples.
The Formula
P(X = k) = C(n,k) × pᵏ × (1-p)ⁿ⁻ᵏ
The binomial distribution gives the probability of exactly k successes in n independent trials, where each trial has the same probability p of success.
Variables
| Symbol | Meaning |
|---|---|
| P(X = k) | Probability of exactly k successes |
| n | Number of trials |
| k | Number of successes desired |
| p | Probability of success on each trial |
| C(n,k) | Binomial coefficient = n! / (k!(n-k)!) |
Example 1
A coin is flipped 10 times. What is P(exactly 7 heads)?
n = 10, k = 7, p = 0.5
C(10,7) = 120
P = 120 × (0.5)⁷ × (0.5)³ = 120 × (0.5)¹⁰
P ≈ 0.1172 ≈ 11.7%
Example 2
A factory has a 3% defect rate. In a batch of 20 items, what is P(exactly 0 defects)?
n = 20, k = 0, p = 0.03
C(20,0) = 1
P = 1 × (0.03)⁰ × (0.97)²⁰ = (0.97)²⁰
P ≈ 0.5438 ≈ 54.4%
When to Use It
Use the binomial distribution when:
- Calculating probabilities for fixed numbers of independent trials
- Quality control — estimating defect probabilities in batches
- Medical studies — probability of treatment success rates
- Any yes/no, pass/fail, or success/failure scenario with fixed trials
Key Notes
- Three conditions must all hold: (1) fixed n trials, (2) each trial is independent, (3) constant probability p on every trial — drawing cards without replacement violates independence (p changes each draw) and requires the hypergeometric distribution instead
- Mean = np, standard deviation = √(np(1−p)) — for 100 fair coin flips, expect 50 ± 5 heads about 68% of the time; these formulas let you quickly assess how "surprising" any observed result is
- Normal approximation: when np ≥ 5 and n(1−p) ≥ 5, the binomial is well-approximated by the normal distribution N(np, np(1−p)) — this enables z-score calculations for binomial problems when n is large without computing large factorials
- Poisson limit: when n is very large and p is very small (with λ = np constant), the binomial approaches the Poisson distribution P(λ) — this is why Poisson models rare events like defects per million or accidents per year