Binomial Probability Calculator
Enter the number of trials, the chance of success each time and the number of successes you are interested in. You get the probability of exactly that many, fewer, more, and the whole distribution.
You can type the chance as a decimal, a fraction such as 1/6, or a percentage such as 25%.
Worked examples
Coin tosses
n = 10, p = 0.5, k = 5
P(exactly 5) = 0.2461
252 ÷ 1,024.
Cumulative
n = 10, p = 0.5, k = 5
P(5 or fewer) = 0.6230
The table running total.
About Binomial Probability Calculator
The binomial distribution describes the number of successes in a fixed number of independent trials, when each trial has the same chance of success. Counting the heads in 10 coin tosses is the classic example. The chance of exactly k successes is nCk × p^k × (1 - p)^(n - k): the number of ways to place the successes, times the chance of each way. For exactly 5 heads in 10 fair tosses that is 252 × (1/2)^10, which is 24.6%.
Questions usually ask for a range rather than one value, such as at least 3 successes. The page works out all of them from the same probabilities: exactly k, fewer than k, k or fewer, more than k and k or more. It also gives the mean, which is n times p, the variance and the standard deviation, and for up to 40 trials it lists the whole distribution with running totals so you can see where most of the probability sits. You can type the chance as a decimal, a fraction like 1/6, or a percentage.
How to use Binomial Probability Calculator
- Enter the number of trials, n.
- Enter the chance of success on each trial as a decimal, a fraction or a percentage.
- Enter k, the number of successes you are interested in.
- Read the probabilities for exactly, at most and at least k, and the table of the whole distribution.
Common uses
- A level statistics questions
- Chances of a number of sixes, heads or defects
- Quality control sampling
- Survey and pass rate predictions
Frequently asked questions
When can I use the binomial distribution?
When there is a fixed number of trials, each trial has two outcomes, the chance of success is the same each time, and the trials are independent.
How do I find the chance of at least one success?
It is 1 minus the chance of none. The page shows k or more directly, and with k = 1 that gives the chance of at least one.
What is the mean of a binomial distribution?
It is n × p, the expected number of successes. For 10 fair coin tosses it is 5.
What is the difference between fewer than k and k or fewer?
Fewer than k stops at k - 1. K or fewer includes k itself. The page shows both so you can match the wording of the question.
Is the answer exact?
It is accurate to around 12 figures. For up to 170 trials the combinations are counted exactly.