The binomcdf function calculates the binomial cumulative distribution (CDF).

Synopsis

binomcdf(x,n,p)

Summary

The binomial distribution represents a trial where there are exactly two mutually exclusive outcomes. These outcomes are labeled success and failure. The binomial CDF gives the probability of obtaining x or fewer successes from n trials with probability of success p. The binomial CDF is:

where I is the indicator function:

Example

Find the Likelihood of Heads-up Coin Tosses

This example finds the likelihood of having a given number of coin tosses turn up heads after 40 coin tosses. The coin is considered fair, that is, the probability of heads in any given toss is 0.5.

Create a 5-by-4 array called heads_array with a double attribute called heads:
```
AFL% CREATE ARRAY heads_array<heads:double>[i=0:4; j=0:3];
```

Put numerical values of 1-20 into the cells of the array:

AFL% store(build(heads_array, (i*4+j)/1.0 + 1.0), heads_array);

Apply the function binomcdf to the values in the attribute heads and store the result in target_array:

AFL% store(apply(heads_array, probability, binomcdf(heads, 20.0, 0.5)), target_array);

The output shows the number of heads along with the cumulative probability of achieving that number of heads:

{i,j} heads,probability
{0,0} 1,2.00272e-05
{0,1} 2,0.000201225
{0,2} 3,0.00128841
{0,3} 4,0.00590897
{1,0} 5,0.0206947
{1,1} 6,0.0576591
{1,2} 7,0.131588
{1,3} 8,0.251722
{2,0} 9,0.411901
{2,1} 10,0.588099
{2,2} 11,0.748278
{2,3} 12,0.868412
{3,0} 13,0.942341
{3,1} 14,0.979305
{3,2} 15,0.994091
{3,3} 16,0.998712
{4,0} 17,0.999799
{4,1} 18,0.99998
{4,2} 19,0.999999
{4,3} 20,1

Thus the chances of getting nine or fewer heads-up results is approximately 41.19%, and of getting 10 or fewer heads-up is about 58.81%. Subtract to compute that the chance of tossing exactly 10 heads is about 17.62%.

Remove the example arrays:

AFL% remove(heads_array); remove(target_array);

Inverse

ibinomcdf: Inverse binomial CDF.