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Jack hears a door being slammed at each interval of six hours, from midnight to 6 a.m. He determines that the sound occurs at an average rate of 13 times per interval. He uses the Poisson distribution to model the number of times he will hear a door slam during any six-hour interval. Keeping this model in mind, what is the probability he will hear a door slam at least once during the one hour from midnight to 1 a.m.?

Answer :

Final answer:

Using the Poisson distribution process, Jack can expect to hear the door slam at least once during the one hour from midnight to 1am with a probability of 88.52%.

Explanation:

The subject of your question falls under the category of Mathematics, particularly statistics. You're looking to apply the Poisson distribution in a practical scenario, which is used to model the number of times an event happens in a fixed interval of time or space.

The average rate of sound occurrence is 13 times in a 6-hour interval. However, to find the probability of the sound occurring in a 1 hour interval, you need to adjust this rate proportionally. So, the 1-hour rate will be 13 (sounds per 6 hours) divided by 6 which is approx 2.1667 sounds per hour.

The probability of hearing a slam at least once is just 1 minus the probability of hearing no slams. The formula for Poisson distribution is P(k; λ) = (λ^k * e^-λ)/k!, where λ is the mean rate (2.1667), k is the number of occurrences, and e is Euler's number (approx 2.71828).

For k=0 (no occurrence), it becomes P(0; 2.1667) = (2.1667^0 * e^-2.1667)/0! = e^-2.1667. After calculating we get approximately 0.1148.

So, the probability of hearing a door slam at least once during the one hour from midnight to 1am is 1-0.1148 = 0.8852, or 88.52%.

Learn more about Poisson Distribution here:

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