How this calculation works
A binomial model requires a fixed number of independent trials with the same success probability. This tool calculates exactly k successes, not k or more. Very small probabilities may underflow to zero.
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)A worked example
Exactly 5 heads in 10 fair coin tosses has probability 24.609375%.
Good to know
Assumes the idealized model described above. Use consistent units and independently verify figures before a consequential decision.
Exactly five is different from at least five
Exactly five successes in ten trials includes outcomes with five successes only. At least five includes five, six, seven, eight, nine and ten. Those are different events and therefore have different probabilities.
The combination term counts the different positions where the successes can occur. One specific sequence of five fair-coin heads and five tails has probability 1/1,024, but there are 252 sequences with exactly five heads, giving 24.609375% in total.
The trials must fit the model
A binomial model assumes a fixed trial count, independence and the same success probability for every trial. Sampling without replacement from a small finite group generally changes the probabilities and can require a different model.
A probability of 0% permits only zero successes; 100% permits only success in every trial. The tool handles those endpoints explicitly. Extremely small probabilities can round or underflow to zero in numerical display.
A few questions, answered
Does this calculate a cumulative probability?
No. It calculates exactly the specified number of successes.
Can successes exceed the number of trials?
No. A trial contributes at most one success in this model.