Binomial and Poisson Distributions

Two discrete distributions

A binomial distribution describes the number of successes in a fixed number of trials. Trials must be independent and share the same success probability. A Poisson distribution models the event count in a fixed time or spatial interval, with expected count λ.

Reading the results

Exact probability applies only to k events. Cumulative probability includes every count from zero through k. The upper tail contains counts strictly larger than k. For five trials with success probability one half, exactly two successes have probability 10/32 = 0.3125. For a Poisson interval with expected count one, the probability of no events is exp(−1). Model choice depends on the structure of the observations.

Distinguish at least from more than

Take four independent trials with success probability 0.5 and threshold two. Exactly two successes have probability C(4, 2) × 0.5⁴ = 6/16 = 0.375. At most two has probability (1 + 4 + 6)/16 = 0.6875. The upper tail includes more than two and equals (4 + 1)/16 = 0.3125. For at least two, add the exact probability to the upper tail, giving 0.6875. The equal probabilities here result from symmetry for four trials and p = 0.5. They are not equal in general.

In the Poisson calculation, λ is the expected total count in the chosen interval, not an event rate. If an hourly rate is known, adjust the expected count for an observation interval of a different duration. In the binomial calculation, p is the probability for one trial, not the observed number of successes.

Calling an event a success does not mean that it is desirable or safe. The same model can represent failures. Do not choose a distribution just because the output looks plausible when trials are dependent or event rates change with time. Report the inequality explicitly for exact, cumulative and upper tail results so that inclusion of the threshold is clear.

Frequently asked questions

How do you calculate a binomial probability?

In this example, for 5 independent trials with success probability 0.5, the binomial probability formula gives 0.3125 for exactly 2 successes. Each trial has the same success probability, and this is an exact count probability rather than the cumulative probability of obtaining at most that many successes.

Calculate with these values