Working with Distributions (CIE A Level Maths: Probability & Statistics 2)

Dominique owns a grocery store, and she models the number of customers entering the store during a 10-minute period using a Poisson distribution with mean 7.

State the assumptions that Dominique has made by using a Poisson distribution to model the number of customers entering her store during a 10-minute period.

Find the probability that no more than 4 customers enter the store during a 10-minute period.

Explain why a normal distribution can be used to approximate the distribution for the number of customers entering the store in a one-hour period. State the appropriate normal distribution.

Using the normal approximation, find the probability that more than 50 customers enter the store within a one-hour period.

In England, it is known that 5% of the population have ginger hair. Kenneth, the owner of a hairdressing salon, has 40 appointments available each day. He models the number of clients with ginger hair that attend his salon in a day using a binomial distribution.

State the assumptions that Kenneth has made by using a binomial distribution.

Assuming all 40 appointments are filled, find the probability that at least one person has ginger hair.

Assuming all 40 appointments are filled each day, the number of clients with ginger hair attending an appointment over a two-day period is denoted . It can be assumed that clients on each day are independent of each other. Explain why a Poisson distribution can be used to approximate . State the appropriate Poisson distribution.

Using the Poisson distribution, find the probability at most 5 clients with ginger hair will have an appointment at the salon over the two-day period.

The random variable is approximated by .

Explain why this approximation is valid and state the value of .

Hence find the percentage error when a Poisson distribution is used to approximate .