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The problem is asking to find the probability of having exactly zero fatal accidents during the next 100 million miles of flight, given that airline fatalities occur at the rate of 0.05 fatal accidents per 100 million miles.

This is a classic Poisson probability problem because we are dealing with a fixed time interval and rare events (fatal accidents). The formula for the Poisson probability is:

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

Where:
- \( \lambda \) is the average number of occurrences in the interval,
- \( k \) is the number of events (accidents) we are interested in,
- \( e \) is Euler's number (approximately 2.71828).

In this case:
- \( \lambda = 0.05 \) (the average number of fatal accidents per 100 million miles),
- \( k = 0 \) (since we are looking for the probability of **zero** fatal accidents).

Substitute the values into the Poisson formula:

\[
P(X = 0) = \frac{0.05^0 e^{-0.05}}{0!} = e^{-0.05}
\]

Now calculate \( e^{-0.05} \):

\[
P(X = 0) = e^{-0.05} \approx 0.9512
\]

Thus, the probability of having exactly zero fatal accidents during the next 100 million miles is approximately **0.9512** (rounded to four decimal places).

Would you like more details on any part of the solution?

Here are some related questions to deepen your understanding:
1. What is the Poisson distribution, and in what situations is it typically used?
2. How does the rate \( \lambda \) affect the probability in a Poisson distribution?
3. What is the difference between the Poisson and binomial distributions?
4. How would the probability change if the rate of fatal accidents increased?
5. How could this probability be applied in real-world decision making for airline safety?

**Tip**: In Poisson problems, remember that the parameter \( \lambda \) represents the expected number of occurrences. If you double the interval, \( \lambda \) also doubles!

According to www.meretrix.com, airline fatalities occur at the rate of 0.05 fatal accidents per 100 million miles. Find the probability that, during the next 100 million miles of flight, there will be exactly zero fatal accidents. Round your answer to four decimal places.

Learn how to calculate the probability of zero fatal accidents over 100 million miles using the Poisson distribution. Given a fatal accident rate of 0.05 per 100 million miles, this problem provides a step-by-step solution to find the probability of no accidents occurring during this interval.

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