Probability of no more than one failure during a particular week is 0.98981.

Poisson Distribution
The Poisson distribution is a probability distribution that denotes the number of events occurring in a fixed interval of time or space. It is often used to model rare events that occur randomly over time.

Given Information
- Average of 3 failures every twenty weeks
- We need to calculate the probability of no more than one failure during a particular week.

Calculating the Rate Parameter (λ)
The average number of failures per week can be calculated by dividing the average number of failures per twenty weeks by the number of weeks in twenty weeks:
λ = (average number of failures) / (number of weeks)
= 3 / 20
= 0.15

Probability of No More than One Failure
To find the probability of no more than one failure during a particular week, we need to calculate the probability of having zero failures and one failure, and then sum them up.

Probability of Zero Failures (P(X = 0))
The probability of having zero failures during a particular week can be calculated using the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!
where:
- P(X = k) is the probability of having k failures
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the rate parameter
- k is the number of failures

For zero failures (k = 0):
P(X = 0) = (e^(-0.15) * 0.15^0) / 0!
= e^(-0.15)
≈ 0.8617

Probability of One Failure (P(X = 1))
Similarly, the probability of having one failure during a particular week can be calculated using the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!

For one failure (k = 1):
P(X = 1) = (e^(-0.15) * 0.15^1) / 1!
= 0.15 * e^(-0.15)
≈ 0.1294

Calculating the Probability of No More than One Failure
To find the probability of no more than one failure, we sum up the probabilities of having zero failures and one failure:
P(X ≤ 1) = P(X = 0) + P(X = 1)
≈ 0.8617 + 0.1294
≈ 0.9911

Therefore, the probability of no more than one failure during a particular week is approximately 0.9911 or 99.11%.