Binomial Probability Calculator
Calculate binomial probabilities P(X = k), P(X ≤ k) and P(X ≥ k), with the mean and standard deviation.
Your inputs
Results
P(X = k)
0.24609375
- P(X ≤ k)
- 0.623046875
- P(X ≥ k)
- 0.623046875
- Mean (np)
- 5
- Standard deviation √(np(1−p))
- 1.581139
✓Exact calculation — rounded only for display
What this calculator does
The binomial distribution models the number of successes in a fixed number of independent trials with the same probability.
Formula and method
P(X = k) = C(n, k) p^k (1 − p)^(n − k), computed in log space for stability
Worked example
Calculated by this tool with its default inputs:
Inputs
- Number of trials (n): 10
- Probability of success (p): 0.5
- Number of successes (k): 5
Results
- P(X = k): 0.24609375
- P(X ≤ k): 0.623046875
- P(X ≥ k): 0.623046875
- Mean (np): 5
- Standard deviation √(np(1−p)): 1.581139
How to read the results
Mean = np; standard deviation = √(np(1 − p)).
Assumptions and limitations
- Trials must be independent with constant probability.
Questions
When does a binomial look normal?
When np and n(1 − p) are both reasonably large (often both ≥ 10).
Methodology and verification
- Result type
- Exact mathematical calculation (rounded only for display)
- Formula version
- 1.0
- Automated verification
- 3 reference and validation test cases, checked against independently computed values. The site cannot be built if any of them fail.
- Editorial review
- Awaiting human editorial review. The explanation was drafted with AI assistance; the formula and examples are verified by the automated tests above.
Privacy
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