THE BINOMIAL DISTRIBUTION

A Binomial distribution arises in situations where there are only two possible outcomes, success or failure. Rolling a die and obtaining a six or not obtaining a six. A component being faulty or not faulty. A marksman hitting a target or not hitting it. Note: The probability of success for each trial should be independent. Examples include:

Example 1: A bag contains a large number of discs. One quarter of the discs are red and the remainder are white. A sample of five discs are selected at random. Find the probability that the sample contains exactly two red discs. The probability of picking red, red, white, white, white in that order is: ×× × × However the number of ways that two red and three white discs can be arranged is 10( or 5 C 2 ). 5 C 2 is the number of ways of arranging 5 objects of which 2 are of one type and 3 are another type. So the required probability is: ×× × × × 10 =

Notation: We write: If X is the r.v. (random variable) “ The number of red discs picked”. X ~ B ( 5, ) 1 4 P(X = 2) = × × C25C2 = In general for a Binomial, the probability of r successes with n trials is given by: P(X = r) = n C r p r (1 – p) n - r

Example 2: 5% of the letters that are sent to a particular office arrive late. Find the probability that of the next 40 letters sent, a) exactly two are late b) no more than four are late c) at least three are late. X ~ B ( 40, 0.05 ) P(X = 2) = × × 40 C 2 = a) ( 4 d.p.) b) P(X ≤ 4) = To calculate this, the above method needs many calculations as above. ( 5 in fact). However there are tables which give the probability directly c) P(X ≥ 3) =1 – P(X ≤ 2) = 1 – =0.3233( 4 d.p.)

Example 3: A survey has shown that 65% of all 10 year old children own a mobile phone. Find the probability that in a random sample of twenty 10 year old children, there are at least 15 who own a mobile phone. If X is the r.v. “ The number of children who own a mobile phone”. X ~ B ( 20, 0.65 ) However, the Binomial tables can not be used as p > 0.5. We want to find P(X ≥ 15) So we need to look at the number of children who do not own a mobile phone. Let Y be the r.v. “ The number of children who do not own a mobile phone”. Y ~ B ( 20, 0.35 ) P(X ≥ 15) = P (Y ≤ 5 ) = Note: At least 15 children owning a mobile phone, is the same as at most 5 not owning one ( Using the tables ).

Summary of key points: This PowerPoint produced by R.Collins ; Updated Nov X ~ B ( n, p ) P(X = r) = n C r p r (1 – p) n - r Means a Binomial distribution with n trials and each trial has probability of success p. For a Binomial, the probability of r successes with n trials is given by: Statistical tables can be used to avoid several calculations. Convert from the number of successes to the number of failures if p > 0.5.