Consider an assembly line in a consumer goods factory that makes facial tissues. Suppose the number of defective facial tissues removed from the assembly line can be approximately modeled using a Poisson process with rate \(\lambda = 3.6\) defective goods per minute.

(a) What is the probability that in any given 45-second period, exactly four defective facial tissues are removed from the assembly line? Round your answer to four decimal places.

(b) What is the probability that the time between two successive removals of defective facial tissues from the assembly line is between 20 and 50 seconds? Round your answer to four decimal places.

(c) Find the mean waiting time between two successive removals of defective facial tissues from the assembly line. Your final answer must be in seconds (not minutes) and accurate to two decimal places.

(a) To find the probability of exactly four defective facial tissues being removed in a 45-second period, we can use the Poisson distribution with a rate (λ) of 3.6 defects per minute. We need to convert the time to minutes to match the rate. The formula for the Poisson distribution is:

P(X = k) = [tex](e^(-λ) * λ^k) / k![/tex]

Where:

P(X = k) is the probability of k events occurring.

e is the base of the natural logarithm (approximately 2.71828).

λ is the rate of events (in this case, 3.6 defective tissues per minute).

k is the number of events (in this case, 4).

Substituting the values into the formula and rounding to four decimal places, we find that the probability is approximately 0.0404.

(b) To find the probability that the time between two successive removals of defective facial tissues is between 20 and 50 seconds, we need to find the cumulative probability of the Poisson distribution with rate λ = 3.6, for the interval [20/60, 50/60] minutes. This involves finding the probability of zero events in the first 20/60 minutes and exactly one event in the next 30/60 minutes, then summing these probabilities. The formula for the Poisson distribution is the same as in part (a).

Performing the calculations and rounding to four decimal places, we find the probability to be approximately 0.1841.

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