QUESTION 6 - 1 POINT
On average, Brendian has noticed that 13 trains pass by his house daily (24 hours) on the nearby train tracks. What is the probability that at most 4 trains will pass his house in a 10 hour time period? (Round your answer to three decimal places.)
Provide your answer below:

Step 1 ：This problem can be solved using the Poisson distribution, which is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Step 2 ：In this case, the mean rate (λ) is 13 trains per 24 hours. However, we are interested in a 10 hour period, so we need to adjust the mean rate accordingly. The adjusted mean rate is \(\frac{13 \times 10}{24} = 5.42\) trains per 10 hours.

Step 3 ：The formula for the Poisson probability is: \(P(x; λ) = \frac{λ^x \times e^{-λ}}{x!}\), where x is the actual number of successes that result from the experiment, e is approximately equal to 2.71828 (Euler's number), and λ is the mean number of successes that occur in a specified region.

Step 4 ：We need to find the probability of at most 4 trains passing by, which means we need to find the sum of the probabilities for 0, 1, 2, 3, and 4 trains.

Step 5 ：The probabilities for 0, 1, 2, 3, and 4 trains are approximately 0.0044, 0.0241, 0.0652, 0.1177, and 0.1593 respectively.

Step 6 ：The sum of these probabilities is approximately 0.371, which is the probability that at most 4 trains will pass his house in a 10 hour time period.

Step 7 ：Final Answer: The probability that at most 4 trains will pass his house in a 10 hour time period is approximately \(\boxed{0.371}\).