Cash Bank wishes to analyse the cost of repairs to ATMs (automated teller machines). The bank

estimates that the probability of a machine developing a fault in the first month is 0.5%. If the

machine does not develop a fault in the first month, the probability of developing a fault in the

second month is 0.8%, whereas if it developed a fault in the first month, the probability of

developing a fault in the second month is 3.0%.

Required:

(a) Calculate the probability of a machine:

(i) not developing a fault during the first two months; and

(ii) developing a fault in either or both of the first two months.

(b) The total number of faults over a year for the bank's portfolio of ATMs can be assumed to

follow a Poisson distribution with a mean equal to 4.5.Calculate the probability that:

(i) no faults are experienced by CASH Bank's ATMs during the year; and

(ii) three or more faults are experienced by CASH Bank's ATMs during the year.

(c) State the assumptions of the Poisson distribution. Discuss two examples of ways in which

the assumptions may not be satisfied in the case of CASH Bank's ATMs.