Binomial Probability Calculator
Binomial Probability Calculator
Enter the number of trials, the probability of success on each individual trial, and the number of successes you're interested in. This calculates both the exact probability of getting precisely that many successes, and the cumulative probability of getting that many or fewer.
The binomial distribution applies whenever you have a fixed number of independent yes/no trials with the same success probability each time — think repeated coin flips, pass/fail quality inspections on a production line, or the number of customers out of 100 who click an ad. It's a workhorse of quality control and A/B-test-style reasoning: manufacturers use it to estimate how many defective units to expect in a batch, and researchers use it to judge whether an observed pass rate is unusually high or low compared to a baseline probability.
- Formula: P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where n is trials, p is success probability, and k is the target successes.
- Requires independent trials: each trial must have the same success probability and not affect the others (like repeated fair coin flips).
- Common uses: quality control (defect rates), A/B test outcomes, quiz/exam guessing probabilities, and any repeated yes/no experiment.
What's the difference between exact and cumulative probability here?
Exact probability is the chance of getting precisely k successes, while cumulative probability is the chance of getting k or fewer successes — cumulative is always the larger (or equal) number.
What counts as a valid "trial" for this calculator?
Any repeatable event with exactly two outcomes (success/failure) and a fixed, constant success probability each time — like flipping a coin, testing a product for defects, or answering true/false questions.
Binomial Probability Calculator
Enter the number of trials, the probability of success on each individual trial, and the number of successes you're interested in. This calculates both the exact probability of getting precisely that many successes, and the cumulative probability of getting that many or fewer.
The binomial distribution applies whenever you have a fixed number of independent yes/no trials with the same success probability each time — think repeated coin flips, pass/fail quality inspections on a production line, or the number of customers out of 100 who click an ad. It's a workhorse of quality control and A/B-test-style reasoning: manufacturers use it to estimate how many defective units to expect in a batch, and researchers use it to judge whether an observed pass rate is unusually high or low compared to a baseline probability.
- Formula: P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where n is trials, p is success probability, and k is the target successes.
- Requires independent trials: each trial must have the same success probability and not affect the others (like repeated fair coin flips).
- Common uses: quality control (defect rates), A/B test outcomes, quiz/exam guessing probabilities, and any repeated yes/no experiment.
What's the difference between exact and cumulative probability here?
Exact probability is the chance of getting precisely k successes, while cumulative probability is the chance of getting k or fewer successes — cumulative is always the larger (or equal) number.
What counts as a valid "trial" for this calculator?
Any repeatable event with exactly two outcomes (success/failure) and a fixed, constant success probability each time — like flipping a coin, testing a product for defects, or answering true/false questions.