The binomial distribution is a discrete distribution, that calculates the probability to get x, the number of successes in an experiment with n trials, and p success probability.
The tool calculates the accumulate distribution function p(x<X), and the inverse distribution function. In the inverse distribution function, there is usually no X that meets the exact probability you enter. The calculator will calculate the X that will generate distribution which is equal or bigger than the input probability but will calculate the probabilities for both X and X-1.
When the binomial calculator can't calculate the distribution or the density (PMF), using the binomial distribution, due to a large sample size and/or a large number of successes, it will use the normal approximation with μ = np and σ=√(np(1-p)).
Binomial probability formula
Following the probability mass function (PMF) formula, and the normal approximation for the binomial distribution.