An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of 4.9 non-work-related emails per hour. Assume the arrival of these emails is approximated by the Poisson distribution.

Required:
What is the probability Linda Lahey, company president, received exactly 3 non-work-related emails between 4 p.m. and 5 p.m. yesterday?

Note: Round your answer to 4 decimal places.

Given that the arrival of non-work-related emails follows a Poisson distribution, we can use the Poisson probability formula to find the probability that Linda Lahey received exactly 3 non-work-related emails between 4 p.m. and 5 p.m. yesterday.

### Poisson Distribution Formula
The Poisson probability formula is given by:

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

Where:
- \( P(X = k) \) is the probability of receiving exactly \( k \) emails.
- \( \lambda \) is the average number of emails received in that time period.
- \( k \) is the number of emails we want the probability for.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

### Given Values
- \( \lambda = 4.9 \) emails per hour (since the time period is 1 hour)
- \( k = 3 \) (we want the probability of receiving exactly 3 emails)

### Calculation
Substituting the values into the formula:

\[
P(X = 3) = \frac{4.9^3 \cdot e^{-4.9}}{3!}
\]

Now, let's calculate this step by step:

1. Calculate \( 4.9^3 \):
   \[
   4.9^3 = 117.649
   \]

2. Calculate \( e^{-4.9} \):
   \[
   e^{-4.9} \approx 0.0074466
   \]

3. Calculate \( 3! \) (which is 3 factorial):
   \[
   3! = 3 \times 2 \times 1 = 6
   \]

4. Now, multiply and divide:
   \[
   P(X = 3) = \frac{117.649 \times 0.0074466}{6} \approx \frac{0.87593}{6} \approx 0.14599
   \]

### Final Answer
The probability that Linda Lahey received exactly 3 non-work-related emails between 4 p.m. and 5 p.m. yesterday is approximately **0.1460** (rounded to 4 decimal places).

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Would you like more details or have any questions?

Here are some related questions:
1. How does the Poisson distribution differ from the normal distribution?
2. What is the probability of receiving at least 3 emails in the same period?
3. How would the probability change if the average number of emails increased to 6 per hour?
4. What are the assumptions underlying the use of the Poisson distribution in this context?
5. How does the Poisson distribution behave when the average rate \( \lambda \) is very high?
6. What is the probability of receiving no emails in an hour?
7. Can the Poisson distribution be used to model other types of events? If so, give examples.
8. How would you calculate the probability of receiving exactly 5 emails?

**Tip:** The Poisson distribution is particularly useful for modeling the number of times an event occurs in a fixed interval of time or space when the events are independent.

Learn how to calculate the probability of receiving exactly 3 non-work-related emails using the Poisson distribution formula. Detailed step-by-step solution with calculations. Suitable for advanced high school students and anyone interested in probability theory.

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