To calculate the probability of certain times between events, we can use Excel:

• [latex]P(\text{at most or less than}) =\text{EXPON.DIST}(x, \lambda, \text{TRUE})[/latex]
• [latex]P(\text{at least or more than}) =1-\text{EXPON.DIST}(x, \lambda, \text{TRUE})[/latex]

In this section, we will revisit the examples from the previous section but use Excel to solve them.

Calculating the probability of AT Most (EXAMPLE)

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: [latex]P(\text{at most or less than}) =\text{EXPON.DIST}(x, \lambda, \text{TRUE})[/latex]

• Time units: minutes
• Lambda: [latex]\lambda = \frac{1}{30 \text{ seconds}} \times \frac{60 \text{ seconds}}{1 \text{ minute}}=2 \text{ } \frac{\text{customers}}{\text{per minute}}[/latex]
• X-values: [latex]P(\text{at most 1}) = P(x \le 1)[/latex]
• Cumulative: TRUE. We always use [latex]\text{Cumulative} = \text{TRUE}[/latex] for exponential distributions. Using [latex]\text{Cumulative} = \text{FALSE}[/latex] 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 [latex]x[/latex]-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 [latex]x[/latex]-value.
• Answer: [latex]P(x \le 1) = \text{EXPON.DIST}(1, 2, \text{TRUE}) = 0.8647[/latex]
• 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.

Calculating the probability of At least (VIDEO)

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:

Applying the Memoryless Property (Exercise)

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.

Key Takeaways (EXERCISE)

Your Own Notes (EXERCISE)

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