What is the probability she received 7 or more non-work-related emails during the same period?

Note: Round your answer to 4 decimal places.



To find the probability that Linda Lahey received 7 or more non-work-related emails between 4 p.m. and 5 p.m. yesterday, we can use the cumulative distribution function (CDF) of the Poisson distribution. The probability of receiving 7 or more emails can be calculated as:

\[
P(X \geq 7) = 1 - P(X \leq 6)
\]

Where \(P(X \leq 6)\) is the cumulative probability of receiving up to 6 emails.

### Poisson Distribution Formula Recap
For each specific number \(k\), the probability \(P(X = k)\) is given by:

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

Where \( \lambda = 4.9 \) (as given).

### Step-by-Step Calculation

1. **Calculate \(P(X = 0)\):**
   \[
   P(X = 0) = \frac{4.9^0 \cdot e^{-4.9}}{0!} = \frac{1 \cdot e^{-4.9}}{1} \approx 0.0074466
   \]

2. **Calculate \(P(X = 1)\):**
   \[
   P(X = 1) = \frac{4.9^1 \cdot e^{-4.9}}{1!} = \frac{4.9 \cdot 0.0074466}{1} \approx 0.0364883
   \]

3. **Calculate \(P(X = 2)\):**
   \[
   P(X = 2) = \frac{4.9^2 \cdot e^{-4.9}}{2!} = \frac{24.01 \cdot 0.0074466}{2} \approx 0.0893956
   \]

4. **Calculate \(P(X = 3)\):**
   \[
   P(X = 3) = \frac{4.9^3 \cdot e^{-4.9}}{3!} \approx 0.1459913
   \]

5. **Calculate \(P(X = 4)\):**
   \[
   P(X = 4) = \frac{4.9^4 \cdot e^{-4.9}}{4!} \approx 0.1783398
   \]

6. **Calculate \(P(X = 5)\):**
   \[
   P(X = 5) = \frac{4.9^5 \cdot e^{-4.9}}{5!} \approx 0.1743719
   \]

7. **Calculate \(P(X = 6)\):**
   \[
   P(X = 6) = \frac{4.9^6 \cdot e^{-4.9}}{6!} \approx 0.1424087
   \]

8. **Calculate \(P(X \leq 6)\) by summing all probabilities from 0 to 6:**
   \[
   P(X \leq 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)
   \]
   \[
   P(X \leq 6) \approx 0.0074466 + 0.0364883 + 0.0893956 + 0.1459913 + 0.1783398 + 0.1743719 + 0.1424087 \approx 0.7744422
   \]

9. **Calculate \(P(X \geq 7)\):**
   \[
   P(X \geq 7) = 1 - P(X \leq 6) \approx 1 - 0.7744422 \approx 0.2255578
   \]

### Final Answer
The probability that Linda Lahey received 7 or more non-work-related emails between 4 p.m. and 5 p.m. yesterday is approximately **0.2256** (rounded to 4 decimal places).

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Would you like more details or have any questions?

Here are some related questions:
1. What is the probability she received exactly 7 non-work-related emails?
2. How would the probability change if the time period was extended to 2 hours?
3. How does the Poisson distribution compare to the binomial distribution for this scenario?
4. What is the expected number of emails she would receive in two hours?
5. How do you calculate the probability for receiving fewer than 5 emails?
6. What is the variance of the number of emails received in this period?
7. How would this probability change if the average rate dropped to 3 emails per hour?
8. What assumptions are made about the emails to justify using the Poisson distribution?

**Tip:** For quick calculations involving the Poisson distribution, many scientific calculators and statistical software can compute cumulative probabilities directly.

Learn how to calculate the probability that Linda Lahey received 7 or more non-work-related emails between 4 p.m. and 5 p.m. using the Poisson distribution and cumulative distribution function. Detailed step-by-step solution with calculations provided.

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