The Binomial Distribution Thursday, 06 December 2018

X ~ Bin (n=4, p= 1/6) Note: q = 5/6 P(X = 0) P(X = 1) Example A die is rolled four times and the number of sixes is recorded. Work out the probability of obtaining 0, 1, 2, 3, or 4 sixes. X ~ Bin (n=4, p= 1/6) Note: q = 5/6 P(X = 0) P(X = 1)

P(X = 2) = = 0.1157 (4 dp) P(X = 3) P(X = 4)

Using either Pascal’s triangle or nCr The above probabilities can all be obtained from the binomial expansion of Using either Pascal’s triangle or nCr P(X=4) P(X=3) P(X=2) P(X=1) P(X=0) Hence this model is known as the Binomial Distribution

Characteristics of a Binomial Distribution A binomial distribution consists of a fixed number of trials (n) where: Each trial has only two outcomes, SUCCESS (p) and FAILURE (q). The probability of failure, q = 1 – p. The outcome of successive trials is independent. X is the random variable such that X is the number of successes.

Example A teacher has a box which contains thirty pencils. With students borrowing pencils the probability that a pencil needs sharpening is 0.2 Find the probability that No pencil needs sharpening Exactly 5 pencils need sharpening No more than two need sharpening

X ~ Bin (n=30, p= 0.2) Note: q = 0.8 a) P(X = 0) b) P(X = 5) c) P(X  2) = P(X = 0) + P(X = 1) + P(X = 2)

X = Number of people that recovered X ~ Bin (n=20, p = 0.4) EXAMPLE The probability that patient recovers from a disease is 0.4. If 20 people contracted the disease, what is the probability that: 13 people exactly recovered. At least 18 people recovered. X = Number of people that recovered X ~ Bin (n=20, p = 0.4) P(X = 13) = = 0.0146 (4 dp)

b) P(X ≥ 18) = P(X = 18) + P(X = 19) + P(X = 20)

Example The proportion of left-handed people in the world is approximately 13%. Based on this proportion, find the probability that in a class of 25 pupils There are no left-handed people There are exactly three left-handed people There is more than one left-handed person.