Binomial Probability Calculator
Exact and cumulative binomial probabilities for independent yes/no trials. Click any i for detail.
Binomial inputs
Trials n
i
n
Number of independent Bernoulli trials (capped at 200 here). |
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|---|---|
Successes k
i
k
Number of successes of interest (0…n). |
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Success probability p
i
p
Probability of success on one trial (0–1). |
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Probability type
i
Type
Exact P(X=k), or cumulative ≤ / ≥. |
How to use this calculator
- Enter n, k, and p.
- Choose exact or cumulative.
- Read the probability.
- Confirm independence and constant p before trusting the model.
Results explained
A binomial random variable counts successes in n independent trials with the same success probability p. The PMF is C(n,k)·p^k·(1−p)^(n−k). Cumulative values sum neighboring PMF terms. Large n may need normal or Poisson approximations in class even when a computer can still sum exactly.
Quick reference: binomial
P(X=k) = C(n,k) p^k (1−p)^(n−k).
| Item | Detail |
|---|---|
| Mean | np |
| Variance | np(1−p) |
| Fair coin, n=10, k=5 | P ≈ 0.246 |
| Assumptions | Independent trials, constant p, two outcomes |
n is capped at 200 on this page for browser-friendly exact sums.
How the estimate is built
P(X=k)=C(n,k)p^k(1−p)^(n−k); cumulative sums PMF terms as selected.
Example scenario
n=10, k=3, p=0.5 → P(X=3) ≈ 0.1172.
FAQ
When is binomial the wrong model?
If trials are dependent, p changes, or there are more than two outcomes, use another distribution.
What about continuity correction?
That appears when you approximate a binomial with a normal — not needed for this exact PMF tool.
Can k exceed n?
No. The page clamps k to n.
Is floating error an issue?
For moderate n the direct product form is fine. Extremely large n needs log-gamma or libraries.