Binomial Probability Calculator
For n independent trials each with success probability p, this calculator gives the chance of exactly k successes — and the cumulative probabilities either side — matching R’s dbinom() and pbinom().
How to use it
The binomial distribution
The binomial distribution counts successes in a fixed number of independent yes/no trials, each with the same probability p — coin flips, pass/fail tests, conversion events. The probability of exactly k successes is:
\[P(X = k) = \binom{n}{k}\,p^{k}(1-p)^{n-k}\]
- P(X = k) — exactly k successes.
- P(X ≤ k) — k or fewer (cumulative); P(X ≥ k) — k or more.
- The distribution has mean np and standard deviation √(np(1−p)).
Use the cumulative versions for questions like “what’s the chance of at least 15 out of 20?” The four assumptions: fixed n, independent trials, two outcomes, constant p.
Do it in R
dbinom(12, size = 20, prob = 0.5) # P(X = 12)
pbinom(12, size = 20, prob = 0.5) # P(X <= 12)
1 - pbinom(11, size = 20, prob = 0.5) # P(X >= 12)FAQ
Frequently asked questions
When can I use the binomial distribution?
When you have a fixed number of independent trials, each with exactly two outcomes (success/failure) and the same success probability. If trials aren’t independent or p changes, the binomial doesn’t apply.
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) sums the probabilities of 0 through k. Use the cumulative form for “at most / at least” questions.
Modelling counts or rare events for real? We can help.