In probability theory and statistics, the binomial distribution is the discrete probability
distribution of the number of successes in a sequence of n independent yes/no
experiments, each of which yields success with probability p.
Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli
trial; when n = 1, the binomial distribution is a Bernoulli distribution.
The binomial distribution is the basis for the popular binomial test of statistical
The binomial distribution is frequently used to model the number of successes in a sample of
size n drawn with replacement from a population of size N.
If the sampling is carried out without replacement, the draws are not independent and so the
resulting distribution is a hypergeometric distribution, not a binomial one.
However, for N much larger than n, the binomial distribution is a good approximation, and
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The following is an example of applying a continuity correction. Suppose one wishes
to calculate Pr(X ≤ 8) for a binomial random variable X.
If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is
approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction; the
uncorrected normal approximation gives considerably less accurate results.
This approximation, known as de Moivre–Laplace theorem, is a huge time-saver
when undertaking calculations by hand (exact calculations with large n are very
it was the first use of the normal distribution, introduced in Abraham de Moivre's book
The Doctrine of Chances in 1738.
Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p)
is a sum of n independent, identically distributed Bernoulli variables with parameter p.
This fact is the basis of a hypothesis test, a "proportion z-test," for the value of p using
x/n, the sample proportion and estimator of p, in a common test statistic.
For example, suppose one randomly samples n people out of a large population and
ask them whether they agree with a certain statement.
The proportion of people who agree will of course depend on the sample. If groups of
n people were sampled repeatedly and truly randomly, the proportions would follow
an approximate normal distribution with mean equal to the true proportion p of
agreement in the population and with standard deviation σ = (p(1 − p)/n)1/2.
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Large sample sizes n are good because the standard deviation, as a proportion of
the expected value, gets smaller, which allows a more precise estimate of the
unknown parameter p.
Suppose individuals with a certain gene have a 0.70 probability of eventually
contracting a certain disease. If 100 individuals with the gene participate in a lifetime
study, then the distribution of the random variable describing the number of
individuals who will contract the disease is distributed B(100,0.7).
Note: The sampling distribution of a count variable is only well-described by the
binomial distribution is cases where the population size is significantly larger than the
sample size. As a general rule, the binomial distribution should not be applied to
observations from a simple random sample (SRS) unless the population size is at
least 10 times larger than the sample size.
To find probabilities from a binomial distribution, one may either calculate them
directly, use a binomial table, or use a computer. The number of sixes rolled by a
single die in 20 rolls has a B(20,1/6) distribution. The probability of rolling more than 2
sixes in 20 rolls, P(X>2), is equal to 1 - P(X<2) = 1 - (P(X=0) + P(X=1) + P(X=2)).
Using the MINITAB command "cdf" with subcommand "binomial n=20 p=0.166667"
gives the cumulative distribution function as follows:
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