Exact and cumulative probabilities for normal, binomial, and Poisson scenarios, each computed with its own formula.
This refreshes immediately whenever you change the distribution or its inputs.
Normal fits continuous, bell-shaped data described by a mean and standard deviation. Binomial fits a fixed number of independent trials with two outcomes each, like pass or fail. Poisson fits counts of events over a fixed interval, described by nothing more than an average rate.
These are the exact formulas behind the binomial and Poisson cases, not something this calculator displays directly but worth knowing when you're checking a result by hand:
Poisson's mean and variance being identical is a useful sanity check: if your real-world counts have a variance much bigger than their mean, a Poisson model may not fit well. See the normal distribution calculator for the continuous case in more depth.
A rough rule of thumb for picking between binomial and its normal cousin: once n × p and n × (1 − p) both climb past about 10, the binomial distribution starts to look enough like a normal curve that either one gives a similar answer. Below that, stick with the exact binomial formula this calculator uses, since the normal shortcut can miss badly for small n or a p far from 0.5.
Because a Poisson process is defined entirely by its average rate, lambda; there's no separate trial count or per-trial success probability to specify. Binomial needs all three (trials, probability, successes) because it's built from repeated independent yes/no trials rather than a single rate.
The exact probability returns 0 automatically. You can't have negative successes or more successes than trials, so the calculator short-circuits those impossible inputs instead of trying to compute a nonsensical factorial.
No. Each distribution is computed with its own exact formula: the normal case uses the cumulative normal curve directly, binomial uses the exact probability mass function, and Poisson uses its own formula. Nothing here approximates one distribution with another.
Binomial, with the success probability set to 0.5 for a fair coin. Set trials to how many flips you're doing and successes to how many heads (or tails) you're asking about; the exact and cumulative probabilities both come out the same way as any other binomial scenario.