The binomial probability distribution

Binomial experiment Many experiments consist of a sequence of n smaller experiments, called trials, that satisfy: Each trial can result in one of two possible outcomes, success (S) or failure (F) The trials are independent The probability of success P(S) is constant from trial to trial. We denote this probability by p. This defines a binomial experiment.

Example 1 The same coin is tossed successively n times. Let S be that the result of a flip is heads. Note that independence follows because P(heads) given any past sequence is ½, the same as on the first trial. The other conditions are satisfied as well.

Example 2 A bag contains 5 red balls and 3 green balls. Two balls are taken successively from the bag, without replacement. Let a success S on a particular trial be that a red ball is drawn. P(S on first trial) = 5/8 P(S on second trial)=(5/8)(4/7)+(3/8)(5/7)=5/8 ! But P(S on second trial given S on first)=4/7 Thus the trials are not independent.

Example 3 A certain state has 500,000 licensed drivers, of whom 400,000 are insured. A sample of 10 drivers is chosen without replacement. The j-th trial is labeled S if the j-th driver chosen is insured. Though this is exactly the same situation as in the last example (sampling without replacement), note what happens because the population size is very large relative to the size of the sample.

Example 3 (continued) The calculations suggest that although the trials are not exactly independent, the conditional probabilities differ so slightly that for practical purposes they can be regarded as being independent.

A rule Consider sampling without replacement from a dichotomous (two types of items) population of size N. If the sample size (number of trials) is at most 5% of the population size, the experiment can be analyzed (probabilities computed) as though it were exactly a binomial experiment.

Definition of a binomial random variable A binomial random variable X associated with a binomial experiment consisting of n trials is defined by X = the number of successes among the trials The distribution of X is denoted by b(x; n, p) in the text to indicate its dependence on the parameters n and p.

Derivation of the distribution of a binomial random variable To compute P(X=x) (the probability of exactly x successes in the n trials), the probability of is . However each of the ways to choose the location of the x successes has exactly the same probability.

Theorem

Mean and variance of X If X~Bin(n,p), then E(X)=np, V(X)= np(1-p), and the standard deviation of X is

Using binomial tables Even for relatively small n, the computation of binomial probabilities can be tedious. Appendix Table A.1 tabulates the cdf for various values of n and p. For a binomial rv,

Example 4 (using Table A.1) A telephone company is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that At most six of the calls involve a fax? .561 Exactly six involve a fax? .561-.378=.183 At least six involve a fax? 1-.378=.622 More than six involve a fax? 1-.561=.439

Example 4 (continued) The expected number of calls among the 25 that involve a fax is E(X)=25(.25)=6.25 The standard deviation of the number of calls among the 25 that involve a fax is The probability that the number of calls that involves a fax exceeds the expected value by more than two standard deviations is

Binomial probabilities Many statistical software packages will provide probabilities associated with binomial distributions. Some websites supply the probabilities as well (for example, see http://stattrek.com/online-calculator/binomial.aspx)