The binomial distribution models yes/no outcomes such as coin flips, defective parts or survey responses.
How to use the Binomial Probability Calculator
- Number of trials (n)
- Successes (k)
- Probability of success
Results update as you type, and every field accepts the units shown in its menu.
How it works
Multiply the number of ways to choose k successes by the probability of each arrangement.
Inputs explained
| Field | What to enter |
|---|---|
| Number of trials (n) | A number |
| Successes (k) | A number |
| Probability of success | In % |
What each result means
- P(X = k) = comb([Number of trials (n)], [Successes (k)])×[p as decimal]^[Successes (k)]×(1 – [p as decimal])^([Number of trials (n)] – [Successes (k)])
- Mean (np) = [Number of trials (n)]×[p as decimal]
- Standard deviation = sqrt([Number of trials (n)]×[p as decimal]×(1 – [p as decimal]))
Example values
How the main result changes when Number of trials changes and everything else stays at the example values above:
| Number of trials (n) | P(X = k) |
|---|---|
| 5 | 0.3125 |
| 7.5 | 0.2486 |
| 10 | 0.1172 |
| 12.5 | 0.0435 |
| 15 | 0.0139 |
| 20 | 0.0011 |
Formula
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏWorked example
Inputs — Number of trials (n): 10; Successes (k): 3; Probability of success: 50 %.
Results — P(X = k) = 0.117188; Mean (np) = 5; Standard deviation = 1.5811.
Frequently asked questions
When can I use the binomial distribution?
Fixed number of independent trials, each with the same success probability.
What is P(3 heads in 10 flips)?
About 0.117.
What formula does the Binomial Probability Calculator use?
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.