8. The random variable has a binomial distribution with unknown index and probability parameter = 0.8. Given that ( = 3) = 8( = 2), find the mean and the variance of the distribution.

9. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.

10. (a) Find the probability that during a 60 minute period, the number of patients arriving at the hospital accident and emergency department is:

(i) exactly 7

(ii) at least 10.

11. An online shop sells a computer game at an average rate of 1 per day. Find the probability that the shop sells more than 10 games in a 7 day period.

12. In a large restaurant an average of 3 out of every 5 customers ask for water with their meal. A random sample of 10 customers is selected. Find the probability of the following events:

a. Exactly 6 ask for water with their meal.

b. Less than 9 ask for water with their meal.

13. A factory produces components of which 1% are defective. The components are packed in boxes of 10. A box is selected at random:

a. Find the probability that the box contains exactly one defective component.

b. Find the probability that there are at least 2 defective components in the box.

c. Use a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.

14. The number of errors made per page by a copy typist has a Poisson distribution with mean 0.2. Calculate the following probabilities:

a. The typist makes exactly 2 errors on a page.

b. The first page typed will contain no error.

c. The typist makes at least 3 errors on a page.

d. Exactly two of the first five pages typed will contain no error.

e. The total number of errors in the first two pages is 2 or less.

15. The random variable X has a Poisson distribution and is such that P(X=2) = 3P(X=4). Find, the correct to three decimal places the values of:

a. ( = 0)

b. ( 4)