Question 11

A certain kind of sheet metal has, on average, 6 defects per 20 square feet.
Assuming a Poisson distribution, find the probability that a 32 square foot metal sheet has at least 12 defects. Round your answer to four decimals.
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Step 1 ：First, we need to find the expected number of defects per 32 square feet. If there are 6 defects per 20 square feet, then the expected number of defects per 32 square feet is calculated as follows: \(\mu = \frac{6 \text{ defects}}{20 \text{ square feet}} \times 32 \text{ square feet} = 9.6 \text{ defects}\)

Step 2 ：We want to find the probability that a 32 square foot metal sheet has at least 12 defects. This is equivalent to finding the probability that the sheet has 12, 13, 14, ... defects and adding these probabilities together. However, it's easier to find the probability that the sheet has 0, 1, 2, ..., 11 defects and subtract this from 1.

Step 3 ：So, we calculate the probability of having 0 to 11 defects as follows: \(P(x \leq 11; 9.6) = P(0; 9.6) + P(1; 9.6) + P(2; 9.6) + ... + P(11; 9.6)\)

Step 4 ：Then, we subtract this from 1 to find \(P(x \geq 12; 9.6)\).

Step 5 ：Let's calculate: \(P(x \leq 11; 9.6) = \sum_{i=0}^{11} P(i; 9.6) = (e^{-9.6}) \times \frac{9.6^0}{0!} + (e^{-9.6}) \times \frac{9.6^1}{1!} + ... + (e^{-9.6}) \times \frac{9.6^{11}}{11!}\)

Step 6 ：After calculating this sum, we subtract it from 1 to find \(P(x \geq 12; 9.6)\). This calculation requires a lot of computation, so it's best to use a calculator or software that can handle Poisson distribution calculations. After calculating, round your answer to four decimals.

Step 7 ：The final answer is \(\boxed{1 - P(x \leq 11; 9.6)}\)