On Monday mornings, the First National Bank only has one teller window open for deposits and withdrawals. Experience has shown that the average number of arriving customers in a four-minute interval on Monday mornings is 2.8, and each teller can serve more than that number efficiently. These random arrivals at this bank on Monday mornings are Poisson distributed.a. What is the probability that on a Monday morning exactly six customers will arrive in a four-minute interval? b. What is the probability that no one will arrive at the bank to make a deposit or withdrawal during a four-minute interval? c. Suppose the teller can serve no more than four customers in any four-minute interval at this window on a Monday morning.What is the probability that, during any given four-minute interval, the teller will be unable to meet the demand? What is the probability that the teller will be able to meet the demand? When demand cannot be met during any given interval, a second window is opened. What percentage of the time will a second window have to be opened? d. What is the probability that exactly three people will arrive at the bank during a two-minute period on Monday mornings to make a deposit or a withdrawal?What is the probability that five or more customers will arrive during an eight-minute period?

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Asked Mar 10, 2020
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On Monday mornings, the First National Bank only has one teller window open for deposits and withdrawals. Experience has shown that the average number of arriving customers in a four-minute interval on Monday mornings is 2.8, and each teller can serve more than that number efficiently. These random arrivals at this bank on Monday mornings are Poisson distributed.



a. What is the probability that on a Monday morning exactly six customers will arrive in a four-minute interval?

 

b. What is the probability that no one will arrive at the bank to make a deposit or withdrawal during a four-minute interval?

 

c. Suppose the teller can serve no more than four customers in any four-minute interval at this window on a Monday morning.What is the probability that, during any given four-minute interval, the teller will be unable to meet the demand?

 

What is the probability that the teller will be able to meet the demand?

 

When demand cannot be met during any given interval, a second window is opened. What percentage of the time will a second window have to be opened?

 

d. What is the probability that exactly three people will arrive at the bank during a two-minute period on Monday mornings to make a deposit or a withdrawal?

What is the probability that five or more customers will arrive during an eight-minute period?
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Expert Answer

Step 1

Hey, since there are multiple subparts posted, we will answer first three subparts. If you want any specific subpart to be answered, then please submit that subpart only or specify the subpart in your message.

Step 2

It is given that there are 2.8 arrivals in a 4-minute interval.

Thus, λ=2.8.

The probability mass function of Poisson distribution is given as follows:

Statistics homework question answer, step 2, image 1

(a)

The probability that exactly 6 customers will arrive can be calculated as follows:

Statistics homework question answer, step 2, image 2

Step 3

(b)

The probability that no one will arrive can be calculated as follows:

Statistics homework question answer, step 3, image 1

(c)

It is given that a teller can serve at most 4 customers in 4-minute interval.

The probability that at most 4 customers will arrive (teller will be able to meet the demand) can be calculated as follows:

Statistics homework question answer, step 3, image 2

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