Chicago has been struck by a crime wave. Alarmed by the increasing number of ban

Question

Chicago has been struck by a crime wave. Alarmed by the increasing number of bank robberies and concerned about their effect on bank customers, the Banking Upper Management Society (BUMS) adopts the following policies at each bank:

• one teller’s window is reserved for the exclusive use of bank robbers.

• in order to conserve space, bank robberies may be committed only by a lone bandit.

• if two or more robberies occur simultaneously, the robbers are served on a first-come, first-serve basis

You are engaged as a consultant by the Bank Robbers Federation (BARF). Your job is to determine if the proposed arrangement with the BUMS is adequate. The data you are given is:

• Robbers arrive at random 24/7 (during open hours), the average arrival rate is 2 robberies per hour.

• Teller service is exponential with an average of 10 min (special robber withdrawal forms expedite service)

determine:

(a) the probability that a bank is hit at least 4 times by robberies during one hour.

(b) the probability that there is no robbery during a whole day.

(c) the probability that the time until the fifth robbery exceeds two days.

(the original version of this problem is due to Shelly Weinberg of IBM)

Explanation / Answer

Solution

Let X = number of robberies in the bank in one hour.

We are given X ~ Poisson (), where is given to be 2 per hour.

Back-up Theory

If a random variable X ~ Poisson(), i.e., X has Poisson Distribution with mean then

probability mass function (pmf) of X is given by P(X = x) = e – .x/(x!) …………..(1)

where x = 0, 1, 2, ……. ,

Values of p(x) for various values of and x can be obtained by using Excel Function.

If X = number of times an event occurs during period t, Y = number of times the same event occurs during period kt, and X ~ Poisson(), then Y ~ Poisson (k) …………….. (2)

Part (a)

Probability that a bank is hit at least 4 times by robberies during one hour

= P(X 4 with = 2 per hour).

Using Excel Function of Poisson Distribution,

= 0.1429 ANSWER

Part (b)

Probability that there is no robbery during a whole day would depend on the number of hours the bank is open. Since this information is not given explicitly in the question, this problem cannot be addressed.

However, taking the number of hours the bank is open to be a variable k, the problem is solved.

Let Y = number of robberies in the bank during a whole day. Then,

By virtue of (2) of Back-up Theory, Y ~ Poisson (2k).

We want P(Y = 0) = e – 2k [by (1) of Back-up Theory].

For k = 6, 7, 8, using Excel Function of Poisson Distribution, the required probabilities are:

Number of hours bank is open

6

7

8

Required Probability

6.14E-06

8.32E-07

1.13E-07

ANSWER

Part (c)

Probability that the time until the fifth robbery exceeds two days, again, would depend on the number of hours the bank is open. Since this information is not given explicitly in the question, this problem cannot be addressed.

However, taking the number of hours the bank is open in a day to be a variable k, the problem is solved.

Let Y = number of robberies in the bank during a period of 2 days. Then,

By virtue of (2) of Back-up Theory, Y ~ Poisson (4k).

Time until the fifth robbery exceeds two days => there is maximum of 4 robberies in 2 days => Y 4. So, required probability   

= P(Y 4)

For k = 6, 7, 8, using Excel Function of Poisson Distribution, the required probabilities are:

Number of hours bank is open

6

7

8

Required Probability

6.21E-07

2.05E-08

6.29E-10

ANSWER

Number of hours bank is open

6

7

8

Required Probability

6.14E-06

8.32E-07

1.13E-07

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