Binomial Probability Calculator

Binomial Probability Calculator gives the chance of a given number of successes in a fixed number of independent trials, each with the same probability. Ten coin flips giving exactly five heads is the textbook case, and the answer is about 24.6 per cent -- noticeably less than most people expect for the most likely single outcome.

All three questions are answered at once, because they are easy to confuse and give very different numbers. Exactly k, at most k and at least k are each useful in different contexts, and the full distribution table shows the whole shape so you can read off any range you need.

The assumptions are where binomial problems go wrong. The trials must be independent and the probability must be the same every time. Flipping a coin satisfies both; drawing cards without replacement satisfies neither, because each draw changes what is left. Applying the binomial formula anyway produces a confident, wrong answer.

The page shows its working, not just its answer. Where there is a formula it is printed with your own numbers substituted in, so you can check the result rather than take it on trust -- which is the difference between a calculator you can use for something that matters and one you cannot.

Everything runs inside your browser. What you type is never uploaded, never logged and never stored, and the calculator keeps working if your connection drops after the page has loaded.

• Shows exactly k, at most k, at least k, expected successes and standard deviation together, so you are not running it twice
• Full the whole distribution you can open and check
• Updates as you type, with no button to press and no page reload
• Copy the result in one click
• Free, no sign-up, and it runs entirely in your browser

The binomial probability calculator is used by writers, developers, students, marketers and anyone else who needs the job done once without installing software. Common cases include:

• Working out the chance of a run of successes
• Checking quality control sampling probabilities
• Finding the probability of at least one failure in a batch
• Teaching or checking a binomial distribution problem