Probability mass function:

Cumulative distribution function:

Mean and variance of a binomial distribution:

Generate a sample of pseudorandom numbers from a binomial distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness:

The distribution is symmetric for :

The distribution becomes symmetric for large n:

Kurtosis:

The limiting value is the value of kurtosis of the standard NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Closed form for symbolic order:

Cumulant:

Hazard function:

Quantile function:

Use dimensionless Quantity to define BinomialDistribution:

CDF of BinomialDistribution is an example of a right-continuous function:

A basketball player has a free-throw percentage of 0.75. Simulate 10 free throws:

Find the probability that the player hits 2 out of 3 free throws in a game:

Find the probability that the player hits the last 2 of 5 free throws:

Find the expected number of hits in a game with n free throws:

A baseball player is a 0.300 hitter. Simulate 5 at bats:

Find the expected number of hits if the player comes to bat 3 times:

A drug has proven to be effective in 30% of cases. Find the probability it is effective in at least 3 of 4 patients:

Find the expected number of successes when applied to 500 cases:

The number of heads in n flips with a fair coin can be modeled with BinomialDistribution:

Show the distribution of heads for 100 coin flips:

Compute the probability that there are between 60 and 80 heads in 100 coin flips:

Now, suppose that for an unfair coin the probability of heads is 0.6:

The distribution and the corresponding probabilities have changed:

A machine produces parts, with 1 in 10 defective:

Compute the probability that at most 1 of 5 parts is defective:

An airplane engine fails with probability p; compute the probability that no more than 2 of 4 fail:

Compute the probability that no more than 1 of 2 fails:

Decide when the choice of four engines is better than two engines:

A system uses triple redundancy with three microprocessors and is designed to operate as long as one processor is still functional. The probability that a microprocessor is still functional after seconds is . Find the probability that the system is still operating after seconds:

With mean time to failure for each processor , find out when the system functions with a probability of less than 99%:

Expressed in years:

Gary Kasparov, chess champion, plays in a tournament simultaneously against 100 amateurs. It has been estimated that he loses about 1% of such games. Find the probability of losing 0, 2, 5, and 10 games:

Use a Poisson approximation to compute the same probabilities:

Perform the same computation when he is playing 5 games, but with stronger opposition so that his loss probability is 10% instead:

In this case the Poisson approximation is less accurate:

A packet consisting of a string of n symbols is transmitted over a noisy channel. Each symbol has probability of incorrect transmission. Find n such that the probability of incorrect packet transmission is less than :

Compute the same limit using a Poisson approximation:

Find the probability that out of n customers need a service if each uses it with probability p:

Compute the probability that more than (capacity) simultaneous service requests are made:

Compute the probability of getting service if and for different capacities :

Find the smallest capacity that provides a 99.9% probability of getting service:

Two players roll dice. If the total of both numbers is less than 10, the second player is paid 4 cents; otherwise the first player is paid 9 cents. Is the game fair?:

The game is not fair, since mean scores per game are not equal:

Find the probability that after n games the player at the disadvantage scores more:

The probability exhibits oscillations:

The maximum of probability is attained at :

Binomial distribution is closed under addition:

BinomialDistribution[n,p] converges to a normal distribution as :

Relationships to other distributions:

BinomialDistribution with is equivalent to BernoulliDistribution:

The sum of n independent variables with BernoulliDistribution is binomial distributed:

BinomialDistribution is the infinite population limit of HypergeometricDistribution:

BinomialDistribution approaches the PoissonDistribution for large n and small p:

A marginal of bivariate multinomial distribution is a binomial distribution:

Confirm by comparing probability mass functions:

BetaBinomialDistribution is a mixture of BinomialDistribution and BetaDistribution:

BinomialDistribution is not defined when p is not between zero and one:

BinomialDistribution is not defined when n is not a positive integer:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: