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)=λke−λk!
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)=25e−25!=32⋅e−2120=32120⋅e2
Calculating this:
P(X=5)≈32120×7.3891≈32886.692≈0.036
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)=64e−64!=1296⋅e−624
Calculating this:
P(X=4)≈129624×403.4288=12969682.2912≈0.134
So, the probability of serving 4 customers in the next 3 minutes is approximately 0.134, or 13.4%.
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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%.