# Binomial distribution Probabilities

Calculates probabilities under a Binomial distribution (including a number of successes between two numbers)

## Binomial distribution formulas

### Notations

X : a random variable following a binomial distribution

n : number of trials

p : probability of success for each trial (number between 0 and 1)

P(X = k) : probability of having exactly k successes in n trials

P(X ≤ k) : probability of having less than k successes in n trials

`P(X = k) = ([n], [k]) * p^k*(1-p)^k`

where `([n], [k])` is the binomial coefficient also called 'n choose k',
`([n], [k]) = (n!)/(k! * (n-k)!)`

These are formulas to calculate lower and upper cumulative binomial distributions,

`P(X <= k) = sum_(i=0)^k ([n], [i]) * p^i*(1-p)^(n-i)`

`P(X >= k) = sum_(i=k)^n ([n], [i]) * p^i*(1-p)^(n-i)`

## See also

Inverse normal distribution Calculator

Statistics Calculators