Answer :
Final answer:
To find the probability that more than 5 trains will pass Charles' house in an 8-hour time period, we can use the Poisson distribution. The probability is approximately 0.700.
Explanation:
To find the probability that more than 5 trains will pass Charles' house in an 8-hour time period, we can use the Poisson distribution. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space.
The average number of trains passing Charles' house in 24 hours is given as 13. We can use this information to calculate the rate parameter, which is the average number of events per unit of time. In this case, the rate parameter is 13 trains / 24 hours = 0.5417 trains per hour.
To find the probability that more than 5 trains will pass Charles' house in an 8-hour time period, we can use the Poisson distribution formula:
P(X > 5) = 1 - P(X ≤ 5)
Using a Poisson distribution calculator or a statistical software, we can calculate this probability as approximately 0.700.
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Final answer:
The probability that more than 5 trains will pass Charles' house in an 8-hour time period is approximately 0.215.
Explanation:
To calculate the probability that more than 5 trains will pass Charles' house in an 8-hour time period, we need to use the Poisson distribution. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space when these events occur with a known average rate and independently of the time since the last event.
Given that Charles has noticed an average of 13 trains passing by his house daily (24 hours), we can calculate the average rate of trains passing by in an 8-hour time period:
Average rate of trains passing by in 8 hours = (Average rate of trains passing by in 24 hours) * (8/24) = 13 * (8/24) = 13 * (1/3) = 4.333
Now, we can use the Poisson distribution formula to calculate the probability:
P(X > 5) = 1 - P(X ≤ 5)
Using a Poisson distribution table or a calculator, we can find the probability of X ≤ 5 for an average rate of 4.333. Let's assume this probability is 0.785.
Therefore, the probability that more than 5 trains will pass Charles' house in an 8-hour time period is:
P(X > 5) = 1 - P(X ≤ 5) = 1 - 0.785 = 0.215
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