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How to use
The binomial distribution models the number of successes in a fixed number of independent trials when the success probability stays constant. For five independent trials with success probability 40%, exactly three successes have probability C(5,3)×0.4³×0.6²=23.04%. At most one success has probability P(0)+P(1)=33.696%. Enter 40 in the percent-probability field, rather than 0.4, and distinguish an exact count from a cumulative probability.
- Enter integer trials n from 0 to 1000.
- Enter success probability p from 0 to 100%, then successes k.
- Calculate P(X=k), P(X≤k), P(X≥k), mean and variance.
How it is calculated
Example
Example: n=10, p=50%, k=5 gives an exact-five probability of about 24.61%.
Important notes
The model assumes independent trials, constant p and two outcomes per trial. Sampling without replacement from a small population may call for a hypergeometric model instead.
Worked examples and interpreting results
Four independent trials with 50% success probability each give probability 37.5% of exactly two successes. Expected successes are 2, not a guaranteed outcome.
| Case | Calculation | Result |
|---|---|---|
| Exactly two successes | 6 × 0.5² × 0.5² × 100 | 37.5 % |
| Expected count | 4 × 0.5 | 2 |
| Exactly three successes: n=5, p=40% | 10 × 0.4³ × 0.6² × 100 | 23.04 % |
| At most one success: n=5, p=40% | (0.6⁵ + 5 × 0.4 × 0.6⁴) × 100 | 33.696 % |
How to check the result
Enter n=4, p=50 and k=2. The model requires fixed trial count, constant probability and independence; sampling without replacement from a small group may need another distribution.
Binomial probability examples
The binomial formula gives the probability of exactly k successes in n independent trials when success probability p is constant for every trial.
- 2 successes out of 4: C(4,2) × 0.5² × 0.5² = 0.375
- Zero successes in 3 trials: 0.8³ = 0.512
- Expected successes in 4 trials: 4 × 0.5 = 2
Important note
The binomial model assumes independent trials with an unchanged success probability; otherwise consider another model.
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Frequently asked questions
Exact versus at most?
Exact k is one outcome count; at most k sums outcomes from 0 through k.
Can p be 0% or 100%?
Yes. The distribution becomes deterministic and the calculator handles both endpoints directly.
When is a binomial model inappropriate?
It may be inappropriate when success probability changes, trials are dependent, or each trial has more than two effective outcomes. Sampling without replacement from a small population may require a hypergeometric model.
How do I calculate the probability of at least k successes?
Use P(X≥k)=1−P(X≤k−1). With n=5 and p=40%, at least three successes has probability 31.744%. Subtracting P(X≤k) would also exclude the required k case.
Is the expected success count a probability?
No. The expected count is np; a particular count has a binomial probability. For n=5 and p=40%, the expected count is 2, but that does not make the probability of two successes equal to 2.