# Documentation

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# binocdf

Binomial cumulative distribution function

## Syntax

`y = binocdf(x,N,p)y = binocdf(x,N,p,'upper')`

## Description

`y = binocdf(x,N,p)` computes a binomial cdf at each of the values in `x` using the corresponding number of trials in `N` and probability of success for each trial in `p`. `x`, `N`, and `p` can be vectors, matrices, or multidimensional arrays that are all the same size. A scalar input is expanded to a constant array with the same dimensions of the other inputs. The values in `N` must all be positive integers, the values in `x` must lie on the interval `[0,N]`, and the values in `p` must lie on the interval [0, 1].

`y = binocdf(x,N,p,'upper')` returns the complement of the binomial cdf at each value in `x`, using an algorithm that more accurately computes the extreme upper tail probabilities.

The binomial cdf for a given value x and a given pair of parameters n and p is

`$y=F\left(x|n,p\right)=\sum _{i=0}^{x}\left(\begin{array}{c}n\\ i\end{array}\right){p}^{i}{\left(1-p\right)}^{\left(n-i\right)}{I}_{\left(0,1,...,n\right)}\left(i\right).$`

The result, y, is the probability of observing up to x successes in n independent trials, where the probability of success in any given trial is p. The indicator function ${I}_{\left(0,1,...,n\right)}\left(i\right)$ ensures that x only adopts values of 0,1,...,n.

## Examples

collapse all

If a baseball team plays 162 games in a season and has a 50-50 chance of winning any game, then the probability of that team winning more than 100 games in a season is:

`1 - binocdf(100,162,0.5)`
```ans = 0.0010```

The result is `0.001` (i.e., `1-0.999`). If a team wins 100 or more games in a season, this result suggests that it is likely that the team's true probability of winning any game is greater than 0.5.