Binomial Distribution Calculator
Compute exact binomial probabilities for any number of trials and success probability, with cumulative values, the full distribution table and a check on whether the normal approximation applies.
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What the binomial distribution describes
It gives the probability of getting exactly k successes in n independent trials when each trial succeeds with the same probability p. Coin flips, defect rates in a production batch, conversion counts in an A/B test and the number of servers failing in a fleet all fit this shape, provided the trials really are independent and p really is constant. Those two assumptions are where the model usually breaks in practice: correlated failures and a drifting rate both violate it, and both make the true distribution wider than the binomial predicts.
Exact beats approximate at small numbers
Textbooks teach the normal approximation because it was necessary before computers. The usual rule of thumb requires np and n(1−p) both above about 5 or 10, and for a rare event with a small p that condition fails no matter how large n is. The exact calculation costs nothing here, so it is used throughout, with the approximation shown alongside so you can see how far off it would have been.
Reading the cumulative values
Almost every practical question is cumulative rather than exact. "What is the chance of at least three failures" and "what is the chance of no more than one defect" are the questions that get asked, while the probability of exactly seven successes is rarely interesting on its own. The exact probability of any single outcome also shrinks as n grows, which surprises people: with a thousand fair coin flips, exactly five hundred heads has a probability of only about 2.5 percent.
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Frequently Asked Questions
When does the binomial distribution not apply?
When the trials are not independent or the success probability is not constant. Correlated server failures, a defect rate that drifts during a production run, and sampling without replacement from a small population all violate the assumptions, and each makes the real spread wider than the binomial predicts.
When is the normal approximation valid?
The usual rule of thumb is that np and n(1−p) should both exceed about 5 to 10. For a rare event with a small p that fails for any n, and the Poisson approximation is the better one. Since the exact calculation is instant here, the approximation is shown only for comparison.
Why is the probability of exactly the expected value so small?
Because the probability spreads over more possible outcomes as n grows. With a thousand fair coin flips the single most likely result is exactly five hundred heads, and its probability is still only about 2.5 percent, since there are a thousand other outcomes sharing the rest.
What is the difference between at least and more than?
At least k includes k itself while more than k does not. Off by one errors here are the most common mistake in binomial problems, which is why every cumulative form is listed separately rather than left for you to derive.
How is this calculated for large n?
Using logarithms of the binomial coefficient, so the intermediate factorials never overflow. Computing the coefficient directly overflows a double precision number at around n of 170, and every value here stays exact well beyond that.
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How to Use
Enter the number of trials, the success probability and the number of successes.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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