Poisson Distribution in Customer Service: Estimating Service Times at the Bank Counter

by Electra Radioti
Poisson Distribution in Customer Service

Exercise

The number of customers served by a bank’s central counter is 2 customers per minute. What is the probability that the counter will:

1. Serve 5 customers in a minute?
2. Serve 4 customers in the next 3 minutes?

Solution

This problem involves a Poisson distribution, as we are dealing with the number of events (customers served) within a fixed period of time.

The Poisson probability mass function is given by:

P(X=k)=λkeλk!

Where:
X is the number of customers served,
λ is the average number of customers served in the given time period,
k is the number of customers we are interested in,
e is the base of the natural logarithm, approximately 2.71828.

In this problem:
– For Question 1, λ=2 (as 2 customers are served per minute),
– For Question 2, λ=6 (as 2 customers per minute over 3 minutes gives 2×3=6).

Question 1: Probability of serving 5 customers in one minute.

We are asked to find P(X=5) for λ=2:

P(X=5)=25e25!=32e2120=32120e2

Calculating this:

P(X=5)32120×7.389132886.6920.036

So, the probability of serving 5 customers in a minute is approximately 0.036, or 3.6%.

Question 2: Probability of serving 4 customers in the next 3 minutes.

We are asked to find P(X=4) for λ=6:

P(X=4)=64e64!=1296e624

Calculating this:

P(X=4)129624×403.4288=12969682.29120.134

So, the probability of serving 4 customers in the next 3 minutes is approximately 0.134, or 13.4%.

Summary of Answers

1. The probability of serving 5 customers in a minute is approximately 3.6%.
2. The probability of serving 4 customers in the next 3 minutes is approximately 13.4%.

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