This problem is based on the Poisson distribution, which is used to model the number of events (breakdowns in this case) occurring within a fixed interval of time, where events happen independently and at a constant rate.

The Poisson distribution is defined by the formula:

$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

Where:

• $\lambda$ is the average number of events (breakdowns) in a given time period.
• $k$ is the number of events.
• $e$ is Euler's number (approximately 2.71828).

In this case:

• $\lambda = 1.5$ (since the process averages 1.5 breakdowns per day).

We want to find the probability of 3 or more breakdowns. This can be calculated by first finding the probabilities of having 0, 1, and 2 breakdowns and then subtracting this sum from 1 (because the total probability must equal 1).

Steps to Calculate in Excel:

1. Calculate probabilities for 0, 1, and 2 breakdowns:

   • Use the Poisson formula in Excel: =POISSON.DIST(k, λ, FALSE)
     • For 0 breakdowns: =POISSON.DIST(0, 1.5, FALSE)
     • For 1 breakdown: =POISSON.DIST(1, 1.5, FALSE)
     • For 2 breakdowns: =POISSON.DIST(2, 1.5, FALSE)

2. Add the probabilities for 0, 1, and 2 breakdowns.

3. Subtract the sum from 1 to get the probability of 3 or more breakdowns: $P(X \geq 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2)$

Example in Excel:

1. For 0 breakdowns:
   =POISSON.DIST(0, 1.5, FALSE) → 0.2231

2. For 1 breakdown:
   =POISSON.DIST(1, 1.5, FALSE) → 0.3346

3. For 2 breakdowns:
   =POISSON.DIST(2, 1.5, FALSE) → 0.2510

4. Total for 0, 1, and 2 breakdowns: $P(X = 0 \text{ or } 1 \text{ or } 2) = 0.2231 + 0.3346 + 0.2510 = 0.8087$

5. The probability of 3 or more breakdowns: $P(X \geq 3) = 1 - 0.8087 = 0.1913$

Thus, the probability of having three or more breakdowns during a day is approximately 0.1913, or 19.13%.

Tip for Excel:

When working with Poisson probabilities, you can also use the cumulative version to directly compute the probability of 2 or fewer breakdowns:
=POISSON.DIST(2, 1.5, TRUE) and then subtract this from 1.

---

Would you like more detailed steps or clarifications on the process?

Here are 5 related questions:

1. What is the probability of exactly 3 breakdowns in a day?
2. How would the probability change if the average breakdowns increased to 2 per day?
3. Can you calculate the probability of fewer than 2 breakdowns using the cumulative function?
4. How would you adjust this method to calculate the probability over a week instead of a day?
5. Can you use a similar method to model rare events in different contexts, like system failures?

Tip: When using the Poisson distribution, the "cumulative" option in Excel saves time by avoiding manual summation.