Enter trials, success probability, and a success count to get exact, cumulative, upper-tail probability, mean, and standard deviation.
Audit PMF/CDF/tails, moments and the highest-probability outcomes.
| k | P(X = k) | CDF | Upper tail |
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Use descriptive context first, choose the method that matches the design, then keep inputs, settings and result evidence reproducible instead of copying a p-value without its analysis recipe.
Reuse n and p from the calculator above, choose the event, and verify the included outcomes on the probability-mass chart.
Use plots and descriptive summaries before formal tests. Check the sampling/design assumptions that matter for the chosen method, report effect size and uncertainty where available, and avoid treating a threshold such as p < .05 as proof of importance, causation, or truth. Browser calculations are educational/planning utilities, not domain-specific professional advice.
A binomial model requires a fixed number of trials, two outcome categories for the event of interest, the same success probability on each trial, and sufficiently independent trials. A correct formula applied to a changing probability or dependent process can still be the wrong model.
Exactly k, at most k, at least k, between two counts, and outside a range describe different sets of outcomes. Use the highlighted PMF bars to confirm that the calculator is summing the outcomes you intended.
The expected count is np and the standard deviation is sqrt(np(1-p)). A requested k far from the mean should usually have a smaller probability than counts near the center, which is a useful reasonableness check.
For at least k, an independent calculation is 1 minus P(X less than k). For outside a closed range, add the lower and upper tails or subtract the between-range probability from 1. These identities are strong checks on cumulative results.
Every outcome from 0 through n can have nonzero probability when 0<p<1. A very small result means the outcome is rare under the stated model, not logically impossible and not automatically evidence of a causal explanation.
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