How to use it
Enter the fixed number of trials n and the probability p for one trial. Choose whether you need exactly k successes, at most k, at least k, or an inclusive interval. For an interval, the first outcome is the lower bound and the final field is the upper bound. The answer appears as a decimal and percentage, followed by the mean, standard deviation, and a compact view of the whole distribution.
For example, with 10 trials and p = 0.5, exactly 3 successes has probability 0.1171875. The interval from 3 through 6 has probability 0.7734375. Changing an input immediately clears the confirmed result so an older answer is never presented as current.
What the calculation means
The binomial model counts successes in repeated trials. Its probability mass for k is the combination count n choose k multiplied by p to the k and (1 − p) to the remaining trials. At-most and interval answers sum the included masses. The calculator evaluates requested tails directly in log space, which avoids factorial overflow and reduces loss of tiny tail values.
The mean n × p is the long-run center, not a promised outcome. The standard deviation √(np(1−p)) describes spread. The bars group outcomes only when n is large; their labels show inclusive outcome ranges and their heights show grouped probability. The requested numeric probability remains the main answer.
Assumptions and edge cases
Trials must be independent, every trial must have exactly two mutually exclusive outcomes, and p must remain fixed. Sampling without replacement from a small population, changing success chances, or dependent events may need another model. The words success and failure are mathematical labels and do not imply value.
n = 0 is allowed and has one possible outcome: zero successes. At p = 0 all mass is at zero; at p = 1 all mass is at n. The limits are computational guardrails, not a statement that larger experiments are invalid. Extremely small probabilities may display in scientific notation.
Examples and interpretation
If a fair coin is flipped 20 times, at least 12 heads means summing outcomes 12 through 20. If a process has p = 0.02 across 100 independent items, at most 1 flagged item means outcomes 0 and 1. Confirm that independence and constant p are defensible before relying on either number.
A probability is not a forecast that the selected count will occur. It quantifies the model under its assumptions. Keep n, p, the requested event, and the result together when copying so a reader can reproduce the calculation.
Frequently asked questions
Does “at least k” include k? Yes. “At most k” also includes k, and an interval includes both endpoints. Can p be entered as 50? No; enter 0.5 for fifty percent. Why are some bars grouped? Showing every outcome for thousands of trials would be unreadable, so the chart groups adjacent outcomes while the numeric answer still uses every integer mass. Is this the same as expanding (a+b)^n? No. The related coefficients appear in the formula, but this page computes probabilities rather than algebraic expansion.