About the Binomial Distribution Calculator
The binomial distribution describes how many times something happens in a fixed number of independent tries, when each try has the same chance of success. Flipping a coin ten times, testing 50 items from a production line or asking 200 voters a yes-or-no question all fit. This calculator takes the number of trials, the chance of success and a number of successes, and gives the probability of exactly that many, the cumulative probabilities for at most and at least that many, and the mean, variance and standard deviation of the whole distribution.
Updated October 2026
How to use the Binomial Distribution Calculator
- 1Enter the number of trials n, such as 10 coin flips.
- 2Enter the chance of success per trial as a decimal like 0.5, or as a percent like 50%.
- 3Enter k, the number of successes you are interested in.
- 4Read the exact probability first, then the 'at most' and 'at least' lines.
- 5Use the mean and standard deviation to judge what counts as typical.
A worked example
A factory finds that 3 percent of its bulbs are faulty. In a box of 40, what is the chance of at least 2 faulty bulbs? Enter n = 40, p = 3% and k = 2. The upper tail P(X ≥ 2) is about 33.8 percent. The mean is 1.2 faulty bulbs per box, so seeing two or more is quite ordinary.
Choosing the right line
- Exactly k: P(X = k).
- At most k, or k or fewer: P(X ≤ k).
- Fewer than k: P(X < k).
- At least k, or k or more: P(X ≥ k).
- More than k: P(X > k).
Related ideas
For a single event see the Probability Calculator, and for counting arrangements try the Permutation and Combination Calculator. The Standard Deviation Calculator and Z-Score Calculator help when n is large enough for a normal approximation, and the Birthday Paradox Calculator shows another classic probability puzzle.
Features
- Exact probability P(X = k)
- Cumulative P(X ≤ k) and P(X < k)
- Upper tail P(X ≥ k) and P(X > k)
- Mean, variance and standard deviation
- Stable for up to 100,000 trials
- Accepts probabilities as decimals or percents
Frequently asked questions
What is the binomial probability formula?
P(X = k) equals n choose k, times p to the power k, times (1 − p) to the power n − k. For 4 heads in 10 fair coin flips that is 210 × 0.5⁴ × 0.5⁶, or 20.5 percent.
When can I use a binomial distribution?
You need a fixed number of trials, each with only two outcomes, the same success chance every time, and trials that don't influence each other. Drawing cards without putting them back into the deck, for instance, breaks the independence condition.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the chance of exactly k successes. P(X ≤ k) adds up the chances of 0, 1, 2 and so on up to k, giving the chance of k or fewer. The calculator shows both, together with the matching upper tail figures.
How do I find the probability of at least k successes?
Use P(X ≥ k), which equals one minus P(X < k). The calculator lists it directly. It saves adding up the long tail by hand when n is large.
What are the mean and standard deviation?
The mean is n times p, the average number of successes you expect. The variance is n times p times (1 − p), and its square root is the standard deviation, which tells you how far results usually stray from the mean.
Does it work for large numbers of trials?
Yes, up to 100,000. The calculation uses logarithms to avoid the overflow that plain factorials would cause, so a result for 5,000 coin flips works fine.