Mathematics and Probability Theory Question 3 (a) The number of calls at a call-centre on average is 5 in every minutes. Calculate the probability that at most 3 calls are received at the call-centre in the period of two minutes. (5 marks) (b) A message is sent through three possible routes with probabilities 0.3, 0.25 and 0.45, respectively. The probability of error through the three routes are 0.1, 0.2 and 0.4, respectively. Suppose that an error free message is received, what is the probability that the message was sent through the first two routes? (5 marks) (c) The time for an individual to be served at a coffee shop is a random variable with an exponential distribution and an average of 3 minutes. Assume that the person goes to the coffee shop every day for 7 days. Calculate the probability that the person is served after 2 minutes on at most two out of those 7 occassions. Clearly define all the random variables and show all your working. (8 marks) (A total of 18 marks for this question.)