A coffee shop Moonbucks classifies its customers into one of the following 3 cat

Question

A coffee shop Moonbucks classifies its customers into one of the following 3 categories: good, average, and bad. Its past records indicate that 35% of its customers are good customers, and 50% are average customers. Further, the amount of time (in hours) that good, average, and bad customers spend in the shop follows an exponential distribution with means of 0.5, 1.0, and 2.0 hours, respectively. Here, the amount of time spent by a customer in the shop is defined to be the time period from the customer buying a drink to the customer leaving the shop. Assume that each customer purchases only one drink. What proportion of Moonbucks' customers stay in the shop for at least 45 minutes after the purchase of their drinks? Given that David, a customer of Moonbucks, bought his coffee at 2:00 PM and left the shop sometime between 3:00 and 3:30 PM, what is the probability that he is in fact a good customer?

Explanation / Answer

required prortion is given by

0.35*e^-3/8+0.5*e^-3/4+0.15*e^-3/2

=0.51

b>.required probability is a conditional probability

=(0.35*e^-1/4)/0.35*e^-1/4+0.5*e^-1/2+0.15*e^-1

=0.38

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