StatReasonWhat this answers
This calculator answers "if I run n independent trials, each with the same success probability, what is the chance of getting exactly (or at least, or at most) k successes?" Classic examples include coin flips, a fixed batch of manufactured parts each with the same defect rate, or a fixed number of survey respondents each with the same probability of a specific response.
When to use it, and when not to
The binomial model requires a genuinely fixed number of trials, a constant success probability across every trial, and independence between trials. If your success probability actually varies from trial to trial, or your sample size itself is random rather than fixed in advance, the binomial model does not apply cleanly.
How it is calculated
The exact probability of k successes in n trials is n-choose-k times p to the k times (1 minus p) to the (n minus k), computed here through a log-gamma-based binomial coefficient for numerical stability at large n. "At least k" and "at most k" sum the exact probabilities of every count in that range using the same stable method, avoiding the precision loss a naive direct sum would suffer for large n.
Worked example
For n=3 trials, p=.5, exactly k=1 success: the probability works out to exactly .375. For the same n and p, at least k=2 successes also works out to exactly .5, since by symmetry of a fair-coin-style binomial the upper and lower halves balance. Try both example inputs above and confirm these exact values.
Assumption audit
Limitations
This calculator returns a probability under the stated binomial model exactly as specified. It cannot tell you whether the binomial model is the right one for your situation, and it does not estimate p from data; for that, use a proportion confidence interval or hypothesis test instead.