A life insurance company has determined that each week an average of eight claims is filed in its Nashville branch. (Round your answers to four decimal places.)What is the probability that during the next week at least sixteen claims will be filed?

To solve this problem, we use the **Poisson distribution**, which is appropriate because it models the number of events (in this case, insurance claims) occurring within a fixed interval of time, given that the events happen independently and at a constant rate.

The Poisson distribution formula is:

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

where:
- \( P(X = k) \) is the probability of exactly \( k \) events occurring.
- \( \lambda \) is the average number of events (claims) per time period (here, per week).
- \( e \approx 2.71828 \) is the base of the natural logarithm.
- \( k! \) is the factorial of \( k \).

Given:
- \( \lambda = 8 \) (the average number of claims filed per week),
- We need the probability that at least 16 claims are filed, i.e., \( P(X \geq 16) \).

This can be computed as:

\[
P(X \geq 16) = 1 - P(X \leq 15)
\]

We will first compute \( P(X \leq 15) \), which is the cumulative probability from 0 to 15, then subtract this value from 1.

Let me calculate that for you.The probability that at least 16 claims will be filed during the next week is approximately **0.0082** (rounded to four decimal places).

Would you like more details on the steps, or do you have any further questions?

Here are five related questions you might find helpful:
1. What is the probability that exactly 16 claims will be filed?
2. How would the probability change if the average number of claims per week increased to 10?
3. What is the probability of filing fewer than 8 claims in a week?
4. Can we apply the Poisson distribution to other types of data? How?
5. How would you calculate the mean and variance in a Poisson distribution?

**Tip:** The Poisson distribution is ideal for modeling rare events over a fixed interval, such as the number of accidents or service requests in a given time period.

A life insurance company has determined that each week an average of eight claims is filed in its Nashville branch. (Round your answers to four decimal places.) What is the probability that during the next week at least sixteen claims will be filed?

Learn how to calculate the probability that at least sixteen claims will be filed in a week using the Poisson distribution. The problem involves a given average of eight claims per week. Detailed solution with step-by-step explanation for Grades 10-12 or college-level students.

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