How to use
- Enter the number of trials n and the probability of success p.
- Under Find, choose exactly k, at most k, fewer than k, at least k, more than k, or between k₁ and k₂.
- Enter k. The result, chart and table update as you type.
Options explained
- Formula
- P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ. The mean is np and the variance np(1 − p).
- When it applies
- Each trial has two outcomes, trials are independent and p is the same every time.
- Exact calculation
- n can be up to 100,000. Terms are computed with the log-gamma function, not the normal approximation, so tail probabilities are accurate.
Worked example
With n = 10 and p = 0.3, the default asks for at most 3 successes:
| Find | Result |
|---|---|
| P(X ≤ 3) | 0.6496 |
| P(X = 3) | 0.2668 |
| Mean np | 3 |
| Variance np(1 − p) | 2.1 |
Other common questions:
| Question | n | p | Find | Probability |
|---|---|---|---|---|
| Exactly 5 heads in 10 tosses | 10 | 0.5 | P(X = 5) | 0.2461 |
| 5% defect rate, at most 2 defects in 10 | 10 | 0.05 | P(X ≤ 2) | 0.9885 |
| Guessing with a 25% chance, at least 7 of 20 right | 20 | 0.25 | P(X ≥ 7) | 0.2142 |
Frequently asked questions
What is the quickest way to get "at least one"?
1 − P(X = 0). Here, choose P(X ≥ k) with k = 1. Five tries at 10% each give 1 − 0.9⁵ = 0.4095.
Can I use the Poisson approximation for large n and small p?
You can, with λ = np, but this calculator handles n up to 100,000 exactly. Open the Poisson calculator to compare.
How does this map to Excel or a TI calculator?
Exactly k is BINOM.DIST(k, n, p, FALSE) or binompdf; at most k is BINOM.DIST(k, n, p, TRUE) or binomcdf.
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