Shylock enters a local branch bank at 430 pm every payday at

Shylock enters a local branch bank at 4:30 p.m. every payday, at which time there are always two tellers on duty. The number X of customers in the bank who are either at a teller window or are waiting in a single line for the next available teller has the following probability distribution.

x: 0, 1, 2, 3, 4, 5, 6

P(x): 0.135, 0.192, 0.284, 0.230, 0.103, 0.051, 0.005

                        a) What number of customers does Shylock most often see in the bank the moment he enters?

Let Y be the number of customers standing in line the moment Shylock enters the bank. First construct the probability distribution of Y . (Hint. What values of X together give Y =0,giveY =1,andsoon?)

                        b) What number of customers waiting in line does Shylock most often see the moment he enters?

                        c) What is the average number of customers who are waiting in line the moment Shylock enters?

Solution

a)

As we see, the number with most probability is x = 3 CUSTOMERS. [ANSWER]

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b)

When there are less than 2 customers, 0 are waiting. Thus,

P(0 waiting) = P(0) + P(1) + P(2) = 0.135+0.192+0.284 = 0.611.

As this is greater than any of the next probabilities P(3), P(4), P(5), P(6), then most often, he sees ZERO (0) CUSTOMERS waiting. [answer]

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C)

Thus,

E(waiting) = 0.609 [ANSWER]