Binomial Probability Calculator: P(X = k), At Most, At Least, Mean and Variance
Enter the number of trials n, successes k and the success probability p. You get exact, cumulative and tail probabilities, the mean and variance, and a bar chart.
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P(X = k)
0.1171875
- P(X ≤ k)
- 0.171875
- P(X ≥ k)
- 0.9453125
- P(X < k)
- 0.0546875
- P(X > k)
- 0.828125
Show working
- P(X = k) = ⁿCₖ · pᵏ · (1 − p)ⁿ⁻ᵏ
- = 10C3 · 0.5^3 · 0.5^7
- 10C3 = 120
- P(X = 3) = 0.1171875
- P(X ≤ 3) = Σ P(X = i), i = 0…3 = 0.171875
- P(X ≥ 3) = 1 − P(X ≤ 2) = 0.9453125
Mean and variance
| Measure | Value |
|---|---|
| μ = np | 5 |
| σ² = np(1 − p) | 2.5 |
| σ | 1.581139 |
How to use
- Enter n (trials) and k (successes).
- Enter p, the chance of success in one trial (0 to 1).
- Read P(X = k), P(X ≤ k), P(X ≥ k) and the chart.
Good to know
Formula: P(X = k) = nCk · pᵏ · (1 − p)ⁿ⁻ᵏ. Example: exactly 3 heads in 10 fair coin tosses = 120 × 0.5¹⁰ = 0.1171875.
Cumulative: P(X ≤ k) adds P(X = 0) … P(X = k); P(X ≥ k) = 1 − P(X ≤ k − 1). Mean μ = np, variance σ² = np(1 − p). The probabilities are computed stably, so n up to 10,000 works.
Use it when trials are independent, each has only two outcomes and p stays the same, e.g. defective items in a batch or correct guesses in a multiple-choice test.
Questions
What is the probability of exactly 3 heads in 10 tosses?
10C3 × 0.5³ × 0.5⁷ = 120/1024 ≈ 0.1172.
What does “at least k” mean?
k or more successes: P(X ≥ k) = P(X = k) + P(X = k + 1) + … + P(X = n).
What are the mean and variance?
Mean np is the average number of successes; variance np(1 − p) measures the spread. For n = 10, p = 0.5 they are 5 and 2.5.
When can I use a normal approximation instead?
Roughly when np and n(1 − p) are both at least 10. This calculator gives exact binomial values, so you do not need to.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.