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:
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]
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