S2Binomial

Distribution

If the probability that James is late home from work on anyday is 0.4, what is the probability that he is late hometwice in a five-day working week?

If there are a large number of stages (or trials), it can bedifficult to rely on Pascal’s triangle

This example is one of a particular type of problem wherethere are only two possible outcomes – a BINOMIALproblem.

The number of paths giving r occurrences out of n cases is:

(𝑛𝑟 )= 𝑛 !𝑟 ! (𝑛−𝑟 )!

nCr =

The BINOMIAL PROBABILITY DISTRIBUTION is defined as:

𝑷 ( 𝑿=𝒓 )=(𝒏𝒓 )𝒑𝒓 (𝟏−𝒑 )𝒏−𝒓 𝒇𝒐𝒓 𝒓=𝟎 ,𝟏 ,𝟐 ,𝟑…𝒏

n stands for the number of trialsp stands for the probability of ‘success’

The particular model can be summarised as:

𝑿 𝑩(𝒏 ,𝒑)

The binomial distribution is a DISCRETE distribution.

If 25 dice are thrown, find the probability that threesixes are obtained.

You can use the binomial distribution for any situation where you want to count the number of times a particularoutcome is observed out of a fixed number of cases –provided certain conditions are satisfied…

There is a fixed number of trials Each trial has the same two possible outcomes The outcomes of the trials are independent of one another The probability of ‘success’ remains constant

You can use standard tables to answer some binomialproblems… 𝑿 𝑩(𝟓 ,𝟎 .𝟑𝟓)What is (a) P(X ≤ 3) (b) P(X = 3) (c) P(X ≥ 3)

If find (a) P(X ≤ 17) (b) P(X > 24)

On 40% if the days that Sean travels to work he finds hehas to stop at a particular set of traffic lights.Find the probability that he has to stop at these lightsno more that five times during a month in which heworks 20 days.

MEAN AND VARIANCE OF THE BINOMIAL DISTRIBUTION

If E(X) = np Var(X) = np(1 – p)

If X ~ B(10, 0.2) find the mean and variance of X

X is a binomial distribution with mean 8 and variance 6.4.Find P(X ≤ 3).

For the following random variables state whether they canbe modelled by a binomial distribution. If they can, give themodel, if they cannot then explain why.(a) A dice is thrown repeatedly until a 1 is seen. X = number of throws.(b) A dice is thrown 10 times. X = number of 1’s seen(c) A bag has 25 red and 25 blue balls in it. Five balls are taken out without replacement. X = number of red balls taken(d) X = number of boys in a family of five children(e) A pair of dice is thrown 25 times. X = number of times a double is thrown(f) A pair of dice is thrown 25 times. X = average score of the sum of the numbers showing.

A vet thinks that the number of male puppies in litters of aGiven size will follow a binomial distribution with p = 0.5.(a) In litters of six puppies, what would be the mean and variance of the number of males if the distribution is binomial?The vet records the number of males in 82 litters of sixPuppies and the results are summarised in the table:

(b) Calculate the mean and variance of the number of males in litters of six puppies.(c) Do you think the binomial distribution is a good model for the number of males in a litter of puppies?

Males 0 1 2 3 4 5 6Frequency 8 10 16 15 14 12 7

If X ~ B(40, p) and Var(X) = 9.6(a) Find the two possible values of p(b) For each of the values of p find P(X < μ – σ)