What is a binomial probability distribution?
A binomial distribution models how many times a “success” happens across a fixed number of independent trials. Each trial has only two outcomes (success/failure), and the probability of success stays the same every time. This calculator helps you find exact and cumulative probabilities quickly.
Where:
- n = number of trials
- k = number of successes
- p = probability of success on one trial
- C(n, k) = number of combinations (“n choose k”)
When should you use the binomial model?
You should use it when all of these are true:
- You run a fixed number of trials.
- Each trial has only two outcomes.
- Trials are independent of each other.
- The success probability is constant.
Examples include coin flips, pass/fail quality checks, and click/no-click outcomes in digital experiments.
How to use this calculator
Step 1: Enter inputs
Type the number of trials n and success probability p.
Step 2: Choose probability type
- P(X = k): exactly k successes
- P(X ≤ k): at most k successes
- P(X ≥ k): at least k successes
- P(a ≤ X ≤ b): between a and b successes, inclusive
Step 3: Click calculate
The tool returns the decimal probability, percentage, and summary statistics (mean, variance, and standard deviation). For smaller values of n, it also shows the full probability table.
Practical interpretation tips
- If P(X = k) is small, that exact outcome is unlikely in one run.
- Cumulative probabilities (≤ or ≥) are often more useful for decision-making.
- The expected number of successes is n × p.
- Spread around that expectation is described by n × p × (1-p).
Common mistakes to avoid
- Using percentages like 40 instead of decimal 0.40.
- Forgetting that k must be an integer between 0 and n.
- Using this model when trial probabilities change over time.
- Using it when outcomes are not independent (without adjustment).
FAQ
Can n be zero?
Yes. With zero trials, there are zero successes with probability 1.
What if p is 0 or 1?
The distribution becomes deterministic: if p = 0, then only zero successes can occur. If p = 1, then exactly n successes occur.
How large can n be?
This calculator can compute large values, but very large n may produce tiny probabilities. In those cases, scientific notation is used for readability.