The Binomial Distribution

The binomial distribution models the number of successes in a fixed number of Bernoulli trials. A Bernoulli trial is an experiment with two possible outcomes, labeled 'success' and 'failure,' where the probability of success has a fixed value for all trials.

The binomial distribution has two parameters: the number of trials, and the probability of success in each trial. If the number of trials is equal to 1, the binomial distribution reduces to the The Bernoulli Distribution.

The binomial distribution arises whenever underlying events have two possible outcomes, and the probability of each outcome occurring remains constant. In general, when a sample of fixed size, n, is taken from an infinite population, and each member of the population independently has a probability p of having a specific property, then the number of members that have this property has a binomial distribution with parameters n and p. If the population is finite, then the same holds, provided the samples were taken independently and with replacement.

Practical examples include:

• The number of times heads or tails is obtained in a fixed number of coin tosses has a binomial distribution.

• The number of defective elements found in a random sample of fixed size from a stable production process has a binomial distribution.

• The binomial distribution can be used to estimate the size of an animal population by marking individuals and releasing them back into the wild.

The binomial distribution is implemented by the BinomialDistribution class. It has two constructors. The first constructor takes one argument: the number of trials. The probability of success is assumed to be 0.5. The following constructs a binomial distribution for 10 trials:

var binomial1 = new BinomialDistribution(10);

Dim binomial1 = New BinomialDistribution(10)

let binomial1 = BinomialDistribution(10)

The second constructor takes two arguments. The first argument is once again the number of trials. The second argument is the probability of success of a trial. The probability must be between 0 and 1. The following constructs a binomial distribution with 10 trials and probability of success 0.4:

var binomial2 = new BinomialDistribution(10, 0.4);

Dim binomial2 = New BinomialDistribution(10, 0.4)

let binomial2 = BinomialDistribution(10, 0.4)

The BinomialDistribution class has two specific properties. NumberOfTrials returns the number of trials. ProbabilityOfSuccess returns the probability of success of a trial.

BinomialDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. It has two overloads, corresponding to each of the two constructors.

var random = new MersenneTwister();
int sample1 = BinomialDistribution.Sample(random, 10);
int sample2 = BinomialDistribution.Sample(random, 10, 0.4);

Dim random = New MersenneTwister()
Dim sample1 = BinomialDistribution.Sample(random, 10)
Dim sample2 = BinomialDistribution.Sample(random, 10, 0.4)

let random = MersenneTwister()
let sample1 = BinomialDistribution.Sample(random, 10)
let sample2 = BinomialDistribution.Sample(random, 10, 0.4)

The above example uses the MersenneTwister class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Distributions.