We will revisit the first example from the previous section:

Example 38.1

Problem Setup: A supermarket records the customer arrivals at the check-outs and finds:

• The time between customer arrivals follows an Exponential distribution.
• The time between customer arrivals is, on average, 30 seconds.

Question: What is the probability that a customer will arrive in the next minute at the check-out?

Solution: Let us use the ‘at most’ formula and follow the steps below:

• Formula: $P(\text{at most or less than}) =\text{EXPON.DIST}(x, \lambda, \text{TRUE})$

• Time units: minutes
• Lambda: $\lambda = \frac{1}{30 \text{ seconds}} \times \frac{60 \text{ seconds}}{1 \text{ minute}}=2 \text{ } \frac{\text{customers}}{\text{per minute}}$
• X-values: $P(\text{at most 1}) = P(x \le 1)$
• Cumulative: TRUE. We always use $\text{Cumulative} = \text{TRUE}$ for exponential distributions. Using $\text{Cumulative} = \text{FALSE}$ gives another function called the PDF (that we do not need in our calculations).
• Complement? No. EXPON.DIST() calculates the probability to the left of (or up to and including) the $x$-value inputted – which is what we are looking for in this example. We only need to take a complement when calculating the area to the right of the $x$-value.
• Answer: $P(x \le 1) = \text{EXPON.DIST}(1, 2, \text{TRUE}) = 0.8647$
• Excel File: Click here to download the Excel file with this solution.

Conclusion: There is an 86.47% chance that a customer will arrive in the next minute.

Next, we will revisit the second example from the previous section:

Example 38.2.1

Problem Setup: The times it takes call center specialist to resolve incoming calls to their call center:

• Follow an exponential distribution.
• Have an average of 2.5 minutes.

Question: What is the probability that a call lasts at least 5 minutes?

Solution: Click here to download the Excel solutions shown in the video below:

Finally, let’s revisit previous example (it’s also the same as the last example from the previous section):

Example 38.2.2

Problem Setup: We will revisit our previous example…

• The times it takes call center specialist to resolve incoming calls follow an exponential distribution.
• On average, it takes the specialist 2.5 minutes to resolve a call.
• After 5 minutes of being on a call, a warning pops up on the specialist’s computer urging them to wrap up the phone call.

Question: What is the probability, after receiving the warning, that it takes at least another 5 minutes to wrap up the call?

You Try: Complete the exercise below to solve the above problem:

Solution: Click here to download the Excel file that contains the solution to the above exercise.

Conclusion: There is 13.53% chance that the call will take at least another 5 minutes to resolve.

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