Solving Telephone Call Probabilities Using Poisson Distribution

When analyzing call data for a car service, you often need to use statistical methods to predict the number of calls in a given period. In this article, we will explore how to use the Poisson distribution to find the probability that the number of calls received in a randomly selected hour is less than 2. Let's dive into the details.

Understanding the Problem

The average number of calls received by a car service is 1.5 calls per hour. This average is denoted by lambda (λ), which in this case equals 1.5 calls/hour. We want to calculate the probability that in a randomly selected hour, the number of calls is less than 2. Mathematically, we need to find the probability that the number of calls is either 0 or 1.

Setting Up the Problem

To solve this problem, we will use the Poisson distribution. The Poisson distribution is used to model the number of events (in this case, telephone calls) occurring in a fixed interval of time or space. The key property of the Poisson distribution is that the average rate of events (λ) is equal to the variance of the distribution.

Using the Poisson Probability Mass Function

The probability mass function (PMF) for a Poisson distribution is given by:

[ P(X k) frac{e^{-lambda} lambda^k}{k!} ]

where ( e ) is the base of the natural logarithm (approximately 2.71828), ( lambda ) is the average rate, and ( k ) is the number of occurrences.

Calculating the Probabilities

We need to find the probability that the number of calls is less than 2, i.e., the sum of the probabilities of receiving 0 or 1 call in a randomly selected hour. This can be written as:

[ P(X

Let's calculate each of these probabilities step by step.

Calculating ( P(X 0) )

The probability that there are 0 calls is calculated as follows:

[ P(X 0) frac{e^{-lambda} lambda^0}{0!} e^{-1.5} cdot frac{1.5^0}{1} e^{-1.5} cdot 1 approx 0.22313 ]

Calculating ( P(X 1) )

The probability that there is 1 call is calculated as follows:

[ P(X 1) frac{e^{-lambda} lambda^1}{1!} e^{-1.5} cdot frac{1.5^1}{1} e^{-1.5} cdot 1.5 approx 0.33469 ]

Combining the Probabilities

To find the combined probability that the number of calls is less than 2, we add the probabilities of receiving 0 and 1 calls:

[ P(X

Thus, the probability that in a randomly selected hour, the number of calls is less than 2 is approximately 0.558, or 55.8%.

Conclusion

Using the Poisson distribution, we can easily calculate the probability of various call scenarios in a defined time frame. In this case, we found the probability of receiving fewer than 2 calls in a randomly selected hour. Understanding these calculations is crucial for optimizing call center operations and providing excellent customer service.