The Binomial Distribution
The Binomial Distribution is used when an event has only two possible outcomes. Within the context of a series of trials these are generally called success and failure. The probabilities of each of these outcomes occurring are denoted by p and q respectively. Since the probability of a success is p, the probability of a failure is therefore . If an experiment is carried out with trials, the probability of successes (and consequently failures) is given by
where N.B. by definition. Because the binomial distribution was originally discovered by Jacob Bernoulli (1654-1705), it is sometimes called the Bernouilli distribution.
i) What is the probability of obtaining 4 heads out of 7 tosses of an unbiased coin?
The tossing of a head is classed as a success. Consequently, the probability of 4 heads is given by
ii) What is the probability of dealing 2 spades if 6 cards are dealt from a normal pack of playing cards?
The probability of dealing a spade is The probability of success (dealing a spade) and failure (not dealing a spade) are respectively and . Consequently, the probability of dealing 2 spades in 6 cards is given by
The mean, , of a binomial distribution is given by .
The variance, , of a binomial distribution is given by .
The standard deviation,, of a binomial distribution is given by .
Find the mean, variance and standard deviation for i) and ii) in the examples given above
i) . Consequently, are given by
ii) . Consequently, are given by
The moment generating function of the binomial distribution is given by