Binomial Probability Calculator
Find the exact and cumulative probability of a number of successes.
How binomial probability works
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success — coin flips, pass/fail quality checks, yes/no survey responses.
Enter the number of trials, the probability of success on each trial, and the number of successes you want to evaluate. The calculator returns both the exact probability of that specific count (PMF) and the probability of that count or fewer (CDF).
This calculator sums the distribution exactly rather than using a normal approximation, so it stays accurate for smaller trial counts and more extreme probabilities where approximations can break down.
The formula
P(X = k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ
C(n,k) is the number of ways to choose k successes from n trials. Multiply by the probability of exactly k successes and n−k failures. The cumulative probability P(X ≤ k) sums this across every count from 0 to k.
Worked example
10 trials, 50% success probability, exactly 5 successes
P(X = 5) = C(10,5) × 0.5⁵ × 0.5⁵ = 252 ÷ 1,024 ≈ 24.6%. P(X ≤ 5) ≈ 62.3%.
Frequently asked questions
What's the difference between PMF and CDF?
PMF is the probability of exactly that many successes. CDF is the probability of that many successes OR FEWER — the running total up to that count.
What assumptions does the binomial model require?
Each trial must be independent with the same success probability throughout — no learning effects, no changing conditions between trials, and a fixed, known number of trials.
When should I use a normal approximation instead?
For very large trial counts this exact calculation remains accurate but can be slower to reason about by hand; a normal approximation (valid roughly when n·p and n·(1−p) are both at least 5–10) is a common alternative for quick estimates.
Next steps in your analysis
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Related guides
- How to Choose the Right Statistical TestA practical framework for picking the right statistical test: start from your research question, data type and study design, then check the assumptions.
- Understanding P-Values and Statistical SignificanceWhat a p-value actually means, what it does not mean, the role of the significance level, and why the field is moving away from the 0.05 bright line.
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