Q1) EXAM PAPER 2018. In a bag, there are 6 marbles. Five are red and one is green. Marbles are randomly selected from the bag and not replaced.

a) What is the probability that the green marble will appear on the first draw? [8]

b) What is the probability that the green marble will appear on the fourth draw? [7]

Assume now that there are 10 marbles in the bag: 5 red, 3 greens and 2 blues. Three marbles are randomly selected.

c) Find the probability that the three marbles are of the same colour if marbles are not replaced after each draw. [6]

d) Find the probability that the three marbles are of the same colour if they are replaced in the bag after each draw. [6]

Q2) The University of Southampton IT service has found that on average 6% of the computers on campus need to be repaired each month. Assume that there are 100 computers on campus.

a) What is the probability that exactly 3 computers will break down this month? [7]

b) Use the Poisson approximation to the binomial distribution to estimate the probability that at least 2 computers will stop working this month [6]

c) Assume now that the number of computer failures during a month has a Poisson distribution with mean 1. A computer has just failed. Find the probability that at least 15 days will elapse before there is a further computer failure (you can assume that there are 30 days in a month). [6]

d) Suppose that 15 days has already passed since the last failure. What is the probability that at least 15 days will elapse before there is another computer failure. [6]

Q3) X is a random variable which follows a Binomial distribution. The mean of the distribution is 20 and the standard deviation is 4. The number of trials is 100.

a) Calculate the probability that X is equal to 15. [6]

b) Calculate the probability that X is equal to 100. [6]

c) What is the probability that X is less than or equal to 62? [6]

d) A researcher chooses a value of X called ‘A’ such that the 20% of the distribution lies below A. What is the value of A? [6]

Q4) A probability density function f(x) is given as follows:

f(x)={█(hx-2h for 2≤x≤3@4h-hx for 3≤x≤4@0 otherwise)┤

a) Find the value of h. [8]

b) What is the mean of the distribution? [4]

c) What is the value of F(2.5)? [4]

d) What is the value of F(1)? [4]

e) What is the value of F(4.5)? [4]