Problem

■Andrew Whittaker, computer center manager, reports that his computer system experienced three component failures during the past 100 days. a. What is the probabifity of no failures in a given day? b. What is the probability of one or more component failures in a given day? c. What is the probability of at least two failures in a 3-day period?

Solution

Step 1 :Calculate the average number of failures per day: \(\frac{3}{100} = 0.03\)

Step 2 :a. Find the probability of no failures in a given day using the Poisson distribution formula: \(P(X=0) = e^{-0.03} \approx 0.9704\)

Step 3 :b. Find the probability of one or more component failures in a given day: \(1 - P(X=0) \approx 1 - 0.9704 = 0.0296\)

Step 4 :c. Calculate the average number of failures in a 3-day period: \(0.03 \times 3 = 0.09\)

Step 5 :Find the probability of at least two failures in a 3-day period: \(1 - P(X=0) - P(X=1) = 1 - e^{-0.09} - (0.09 \times e^{-0.09}) \approx 0.0038\)

Step 6 :\boxed{\text{Final Answer:}}

Step 7 :a. \(\boxed{0.9704}\) is the probability of no failures in a given day.

Step 8 :b. \(\boxed{0.0296}\) is the probability of one or more component failures in a given day.

Step 9 :c. \(\boxed{0.0038}\) is the probability of at least two failures in a 3-day period.

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Source: https://solvelyapp.com/problems/8362/

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