About the Binomial Distribution Calculator
What is the Binomial Distribution Calculator?
The binomial distribution gives the probability of getting exactly k successes in n independent trials, when each trial has the same probability p of success — for example, the number of heads in 10 coin tosses or defective items in a batch of 50.
Enter n, k and p (as a percentage). The calculator returns the exact probability P(X = k), the cumulative probabilities P(X ≤ k) and P(X ≥ k), and the mean and standard deviation of the number of successes.
How to use the Binomial Distribution Calculator
- Number of Trials (n) (sample: 10)
- Number of Successes (k) (sample: 3)
- Probability of Success per Trial (%) — the success probability per trial, as a percentage (50 for a fair coin). (sample: 50)
- Press Calculate Probability (results also refresh as you type). The result card shows the P(X = k), along with P(X ≤ k), P(X ≥ k), Mean (np) and Standard deviation √(np(1−p)).
Formula used
P(X = k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ
- n = the number of trials
- k = the number of successes
- p = the probability of success on each trial
- nCk = the number of ways to choose which k trials succeed
Mean = np and standard deviation = √(np(1 − p)). The calculator works with logarithms of factorials, so large n (up to 1,00,000) does not overflow.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Number of Trials (n) | 10 |
| Number of Successes (k) | 3 |
| Probability of Success per Trial (%) | 50 |
| Result | Value |
|---|---|
| P(X = k) | 0.117188 |
| P(X ≤ k) | 0.171875 |
| P(X ≥ k) | 0.945313 |
| Mean (np) | 5 |
| Standard deviation √(np(1−p)) | 1.5811 |
Tips for using the Binomial Distribution Calculator
- The trials must be independent with a constant p — drawing cards without replacement is not binomial.
- “At least k” is P(X ≥ k); “at most k” is P(X ≤ k).
- For large n, the binomial is close to a normal distribution with mean np and SD √(np(1 − p)).