Binomial Distribution Formula
Calculate the probability of getting exactly k successes in n independent trials.
Used in quality control and experiments.
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