Binomial Probability Calculator
Calculate the exact, cumulative, and complementary probabilities for a binomial distribution given the number of trials, success probability, and target number of successes.
P(X = 5) - exact probability
24.6094%
P(X ≤ 5) - at most k successes
62.3047%
P(X ≥ 5) - at least k successes
62.3047%
Expected value (np)
5.000
Standard deviation
1.581
Recommended Settings
Pro Tips
- •The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success
- •P(X = k) is the exact probability of getting exactly k successes, while P(X ≤ k) and P(X ≥ k) give you the cumulative probability across a range of outcomes
- •The expected value (n × p) tells you the average number of successes you'd see if you repeated the whole experiment many times
- •This tool supports up to 1,000 trials using a numerically stable calculation that avoids overflow from extremely large factorials
Most Popular
Most students test small values like n=10, p=0.5 to check textbook examples before applying the tool to their actual problem
When to Use This Tool
Check binomial probability calculations for coursework or exam practice.
Estimate the probability of a certain number of defective items in a production batch.
Calculate the odds of a specific number of successes across repeated independent events, like coin flips or A/B test conversions.
Double-check a binomial probability you calculated by hand or with another tool.
How It Works
Compute the binomial coefficient (n choose k) using a numerically stable multiplicative method that avoids overflow for large n
Multiply by p raised to the k successes and (1-p) raised to the remaining failures to get the exact probability
Sum the exact probabilities across the relevant range of outcomes to compute cumulative probabilities
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Open Source
Built with verified, open-source libraries. Fully transparent.
Frequently Asked Questions
What conditions does the binomial distribution require?
It requires a fixed number of independent trials, each with only two possible outcomes (success or failure), and the same probability of success on every trial.
What's the difference between P(X=k) and P(X≤k)?
P(X=k) is the probability of exactly k successes. P(X≤k) is the cumulative probability of getting k or fewer successes, summing the exact probabilities for every outcome from 0 up to k.
Why is there a limit of 1,000 trials?
Beyond that range, the numbers involved become extremely large or small, and while the calculation method remains accurate, results become less practically meaningful for typical use cases.
Is my data sent anywhere?
No. All calculations happen locally in your browser.