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Children arrive at a house to do Halloween trick-or-treating according to a Poisson process at the rate of 13 per hour. What is the probability that the time between the 15th and 16th arrivals will be more than 4 minutes? (Hint: Think exponential.)

Answer :

Final answer:

The probability that the time between the 15th and 16th trick-or-treaters will be more than 4 minutes is approximately 0.4956.

Explanation:

To solve this problem, we can use the exponential distribution, which is a continuous probability distribution that describes the time between events in a Poisson process. The average rate of arrival for the trick-or-treaters is 13 per hour, so we can calculate the rate parameter λ (lambda) by dividing the rate by 60 (to convert it to minutes): λ = 13/60 = 0.2167. The probability of the time between the 15th and 16th arrivals being more than 4 minutes can be calculated using the cumulative distribution function (CDF) for the exponential distribution. We subtract the CDF value at 4 minutes from 1 to find the probability of the time being more than 4 minutes: P(X > 4) = 1 - F(4), where F(x) is the CDF for the exponential distribution. Using the formula for the exponential CDF, F(x) = 1 - e^(-λx), we can substitute the values to find the probability: P(X > 4) = 1 - e^(-0.2167*4) = 1 - e^(-0.8668) ≈ 0.4956.

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