Binomial Probability Calculator
Calculate the probability of exactly k successes out of n independent trials, using the binomial distribution.
When the binomial distribution applies
The binomial distribution models situations with a fixed number of independent trials, each with only two possible outcomes (success or failure), and the same probability of success on every trial — classic examples include counting heads in a series of coin flips, or counting defective items in a batch where each item has the same defect probability.
If trials aren't independent (the outcome of one affects the next) or the success probability changes between trials, the binomial distribution doesn't apply and a different probability model would be needed.
Interpreting the result
This calculator gives the probability of exactly k successes, not 'k or more' or 'k or fewer' — for cumulative probabilities across a range of outcomes, you would need to sum the individual probabilities for each value in that range.
The expected value (mean) tells you the average number of successes you'd expect over many repetitions of this same n-trial experiment, while variance measures how much the actual results tend to spread out around that average.
Frequently asked questions
What's required for the binomial distribution to apply?
A fixed number of independent trials, each with exactly two possible outcomes, and the same success probability on every trial — if any of these conditions aren't met, a different probability model is needed.
Does this give the probability of exactly k successes, or k or more?
Exactly k successes. For cumulative probabilities (like 'k or more'), you would need to sum this calculation across each relevant value of k.