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Answer :
To solve the problem of determining the probability that at most 4 trains will pass Brendan's house in a 10-hour period, we can use the Poisson distribution. The Poisson distribution is useful for predicting the number of events occurring within a fixed interval of time, given a known average rate of occurrence.
Here's a step-by-step breakdown of the solution:
1. Determine the Average Rate for 10 Hours:
- We know that, on average, 13 trains pass by Brendan's house in a 24-hour period.
- To find the average number of trains passing in a 10-hour period, we scale this average rate accordingly:
[tex]\[
\text{Average number of trains in 10 hours} = 13 \times \left(\frac{10}{24}\right) = 5.417
\][/tex]
2. Use the Poisson Distribution:
- The Poisson distribution is characterized by a parameter [tex]\(\lambda\)[/tex], which is the average rate of occurrence over the interval. Here, [tex]\(\lambda = 5.417\)[/tex].
- We want to find the probability that at most 4 trains pass by in 10 hours. This is calculated as the cumulative probability of 0, 1, 2, 3, or 4 trains passing.
3. Cumulative Probability Calculation:
- The probability of at most [tex]\( k \)[/tex] occurrences in a Poisson distribution is given by the cumulative distribution function (CDF). So, we calculate the CDF at [tex]\( k = 4 \)[/tex].
4. Final Result:
- The cumulative probability of at most 4 trains passing in 10 hours is approximately 0.371.
This means there is about a 37.1% chance that 4 or fewer trains will pass Brendan's house in the 10-hour period.
Here's a step-by-step breakdown of the solution:
1. Determine the Average Rate for 10 Hours:
- We know that, on average, 13 trains pass by Brendan's house in a 24-hour period.
- To find the average number of trains passing in a 10-hour period, we scale this average rate accordingly:
[tex]\[
\text{Average number of trains in 10 hours} = 13 \times \left(\frac{10}{24}\right) = 5.417
\][/tex]
2. Use the Poisson Distribution:
- The Poisson distribution is characterized by a parameter [tex]\(\lambda\)[/tex], which is the average rate of occurrence over the interval. Here, [tex]\(\lambda = 5.417\)[/tex].
- We want to find the probability that at most 4 trains pass by in 10 hours. This is calculated as the cumulative probability of 0, 1, 2, 3, or 4 trains passing.
3. Cumulative Probability Calculation:
- The probability of at most [tex]\( k \)[/tex] occurrences in a Poisson distribution is given by the cumulative distribution function (CDF). So, we calculate the CDF at [tex]\( k = 4 \)[/tex].
4. Final Result:
- The cumulative probability of at most 4 trains passing in 10 hours is approximately 0.371.
This means there is about a 37.1% chance that 4 or fewer trains will pass Brendan's house in the 10-hour period.
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