Binomial Distribution Calculator
Use the binomial distribution when you repeat the same yes/no trial n times, each with the same chance of success p. Enter n, p and the number of successes x to get exact (PDF) and cumulative (CDF) probabilities.
Results update as you type.
- P(X = 6)
- 0.205078
- P(X ≤ 6)
- 0.828125
- P(X < 6)
- 0.623047
- P(X ≥ 6)
- 0.376953
- P(X > 6)
- 0.171875
- Mean (np)
- 5
- Standard deviation
- 1.5811
Steps
- P(X = 6) = C(10, 6) × 0.5^6 × (1 − 0.5)^4 = 0.205078
Worked example
Example: exactly 6 heads in 10 fair coin flips
- C(10, 6) = 210
- 0.5⁶ × 0.5⁴ = 0.0009765625
- 210 × 0.0009765625 = 0.205078
P(X = 6) ≈ 0.2051 (20.5%).
Formula
P(X = x) = C(n, x) · pˣ · (1 − p)ⁿ⁻ˣ
Mean = np, SD = √(np(1 − p))
Cumulative probabilities add the individual terms: P(X ≤ x) = Σ P(X = k) for k = 0 … x.
Frequently asked questions
What are the conditions for a binomial experiment?
A fixed number of trials, two outcomes per trial, independent trials and the same probability of success each time.
What's the difference between binompdf and binomcdf?
binompdf gives P(X = x); binomcdf gives P(X ≤ x).
When can I use a normal approximation?
A rule of thumb is when np ≥ 10 and n(1 − p) ≥ 10.
Last reviewed October 17, 2026 · Calculations run in your browser; nothing you enter is sent to a server.