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📘 About the Binomial Distribution Calculator
The binomial distribution counts successes in a fixed number of independent trials, each with the same chance of success — heads in coin tosses, correct guesses on a multiple-choice test, defective items in a sample, patients responding to a treatment.
Enter n, p and k and choose exactly, at most, at least or between. The calculator computes the exact probability (no normal approximation), the cumulative probabilities, the mean np, variance np(1 − p), the most likely number of successes and a chart of the whole distribution.
🧮 Formula
✏️ Worked examples
6 heads in 10 tosses
You get| P(X = 6) | 0.205078 = 20.5078% |
| P(X = 6) | 0.205078 |
| P(X ≤ 6) · P(X ≥ 6) | 0.828125 · 0.376953 |
| Mean np | 5 |
| Variance np(1 − p) · SD | 2.5 · 1.5811388 |
| Most likely number of successes | 5 |
| Formula | P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ = C(10, 6)·0.5^6·0.5^4 |
0.205078 = 20.5078%.
At least 8 of 20 at p = 0.3
You get| P(X ≥ 8) | 0.227728 = 22.7728% |
| P(X = 8) | 0.114397 |
| P(X ≤ 8) · P(X ≥ 8) | 0.886669 · 0.227728 |
| Mean np | 6 |
| Variance np(1 − p) · SD | 4.2 · 2.0493902 |
| Most likely number of successes | 6 |
| Formula | P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ = C(20, 8)·0.3^8·0.7^12 |
0.227728 = 22.7728%.
🧭 How to use this calculator
What you enter
- Number of trials n
- Probability of success p0.5, or 50 for 50%
- Number of successes k
- Probability of
- k₂ (for between)
What you get
- P(X = 6)
- P(X = 6)
- P(X ≤ 6) · P(X ≥ 6)
- Mean np
- Variance np(1 − p) · SD
- Most likely number of successes
- Formula
❓ Frequently asked questions
When is a situation binomial?
When there is a fixed number of trials, each with two outcomes, the same probability of success, and the trials are independent.
What is the probability of guessing 5 of 10 multiple-choice questions right (4 options)?
P(X = 5) with n = 10, p = 0.25 is about 5.8%; at least 5 is about 7.8%.
How is “at least” calculated?
P(X ≥ k) = 1 − P(X ≤ k − 1), adding up the probabilities of every lower count.
