When to use the binomial distribution
The binomial distribution counts successes in a fixed number of independent trials that each have the same chance of success. It answers questions like: what is the chance that exactly 7 of 20 customers buy, that at least 3 of 50 parts are defective, or that a basketball player makes at most 6 of 10 free throws? Four conditions must hold: a fixed number of trials n, two outcomes per trial, the same probability p each time, and independent trials.
The formulas
- P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), with C(n, k) = n! ÷ (k! × (n − k)!)
- P(X ≤ k): the sum of P(X = i) for i from 0 to k
- P(X ≥ k) = 1 − P(X ≤ k − 1)
- Mean = n × p; standard deviation = √(n × p × (1 − p))
The calculator sums the exact probabilities, so it stays accurate even for large n, where hand calculation is impractical.
Worked example
A sales rep closes 30% of the qualified calls they make. Out of 20 calls this week, what is the chance of closing 7?
- P(X = 7) = C(20, 7) × 0.3⁷ × 0.7¹³ = 77,520 × 0.0002187 × 0.009689 = 16.4%.
- P(X ≤ 7) = 77.2%: most weeks bring 7 sales or fewer.
- P(X ≥ 7) = 39.2%: a week of 7 or more sales happens about two times in five.
- On average the rep closes 6 sales, with a standard deviation of 2.05.
The full distribution
For n = 20 and p = 30%:
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|---|---|---|
| 2 | 2.8% | 3.5% | 99.2% |
| 4 | 13.0% | 23.8% | 89.3% |
| 5 | 17.9% | 41.6% | 76.2% |
| 6 | 19.2% | 60.8% | 58.4% |
| 7 | 16.4% | 77.2% | 39.2% |
| 8 | 11.4% | 88.7% | 22.8% |
| 10 | 3.1% | 98.3% | 4.8% |
| 12 | 0.4% | 99.9% | 0.5% |
The most likely single outcome is 6, the mean, but even it only happens 19% of the time.
Related distributions
- Normal approximation: when n × p and n × (1 − p) are both above about 10, the binomial is close to a normal distribution with the same mean and standard deviation.
- Poisson: for rare events, with large n and small p, the binomial approaches a Poisson distribution with mean n × p.
- Hypergeometric: when you draw without replacement from a small population, such as cards from a deck, the trials are not independent and the binomial overestimates the spread.
For coin tosses specifically, the coin flip calculator uses the same mathematics with p = 50% by default.