The binomial distribution covers any fixed number of independent trials with the same success probability each time — coin flips, free throws, defective parts on a production line. The formula multiplies the chance of one particular sequence with k successes by the number of sequences that contain k successes, which is the combination count.
The cumulative figures are usually what a real question needs. "At least three defects in a batch of twenty" is a sum over several exact probabilities, and this page computes it directly. The mean is n times p and the standard deviation is the square root of np(1−p), which for large n makes the distribution close enough to normal that the z-score approach works as a shortcut.
The exactly k heads in twenty fair coin flips table follows the calculator: 9 rows spanning 0 through 20, there so the result can be cross-checked against a value you already trust. Nothing has to be entered to see it work: the default for number of trials is 20, and the output recomputes the moment that changes. The 4 bands (very unlikely at one end, most likely region at the other) exist to answer the question a number on its own cannot: is this high, low or ordinary? The pages people most often open next are Normal Distribution Calculator and Birthday Problem Calculator.
The formula
n- Number of trials
k- Number of successes
p- Success probability per trial
C(n, k)- Number of ways to arrange k successes
How it works, step by step
- Enter the number of trials.
- Enter the number of successes you are asking about.
- Set the success probability for a single trial.
- The gauge shows the chance of exactly that many successes.
Worked examples
Twelve heads in twenty flips
C(20, 12) = 125,970, and each specific sequence has probability 0.5 to the twentieth. The result is 12.0134%, against 17.6197% for the most likely outcome of ten.
Cumulative rather than exact
The chance of twelve or more heads in twenty flips is 25.1722%, considerably higher than the 12.0134% for exactly twelve.
Checking the 10 row
The reference table lists k = 10, giving Ways C(20, k): 184,756, Exactly k: 17.6197%, k or more: 58.8099%. Entering that above should reproduce the same figures — if it does not, the input is being read differently to the way the table assumes.
How to read your score
Frequently asked questions
What conditions does the binomial distribution require?
A fixed number of trials, two possible outcomes each, a constant success probability, and independence between trials. Break any of these and the formula no longer applies.
Why is the exact probability often so small?
Because it spreads across many possible counts. With twenty trials there are twenty-one possible outcomes, so even the most likely one rarely exceeds twenty per cent.
Should I use exact or cumulative probability?
Cumulative for almost every real question. "At least" and "no more than" are what quality control and hypothesis testing actually ask.
When can I approximate with a normal distribution?
When np and n(1−p) both exceed about five. The engine reports the mean and standard deviation you would need for that approximation.
What do the bands under the chance of exactly k mean?
They label the figure so it can be read without a reference point of your own: Very unlikely from 0 to 1; Unlikely from 1 to 6; Plausible from 6 to 15; Most likely region from 15 upwards. The band is a reading aid only — the number itself is exact, and a value sitting just over a boundary is not meaningfully different from one just under it.
What formula does the Binomial Probability Calculator use?
It computes P(k) = C(n, k) p^k(1 − p)^n−k, where n is number of trials; k is number of successes; p is success probability per trial; C(n, k) is number of ways to arrange k successes. The same expression is printed on the page above the inputs, so the result can be reproduced by hand or in a spreadsheet.
Binomial probabilities for twenty trials
| k | Ways C(20, k) | Exactly k | k or more |
|---|---|---|---|
| 0 | 1 | 0.0001% | 100.0000% |
| 4 | 4,845 | 0.4621% | 99.8712% |
| 6 | 38,760 | 3.6964% | 97.9305% |
| 8 | 125,970 | 12.0134% | 86.8412% |
| 10 | 184,756 | 17.6197% | 58.8099% |
| 12 | 125,970 | 12.0134% | 25.1722% |
| 14 | 38,760 | 3.6964% | 5.7659% |
| 16 | 4,845 | 0.4621% | 0.5909% |
| 20 | 1 | 0.0001% | 0.0001% |
The distribution is symmetric because the success probability is exactly one half.