a manufacturing process leaves an average of 3 defects per 10m^2 of manufactured

Question

a manufacturing process leaves an average of 3 defects per 10m^2 of manufactured product. the number of defects follows a Poisson distribution.

a)what is the probability that a 4 x 5m product will contain more than 2 defects?

b)what is the probability that a 4 x 6m product will contain more than 2 defects?

c)what is the probability that 50 products (each 4 x 10m) will contain more than 300 defects total?

Explanation / Answer

The mean number of defects per squ. metre is 3/10 = 0.3. a) If the area is 4m by 5m = 20 squ. metres, then the mean is 20 x 0.3 =6. So X, the number of defects, is Po(6) and P(X=x) = exp(-6) x 6^x/x!. P(X>2) = 1- P(X=0) - P(X=1) - P(X=2) = 1 - 0.0024788 = 0.0148725 - 0.044618 = 0.938 = 0.94 (2 dp) b) If the areas is 4m by 6m, then the mean is 24 x 0.3 = 7.2. So X is Po(7.2). P(X>2) = 1- P(X=0) - P(X=1) - P(X=2) = 1- 0.0007466 - 0.005375 - 0.0193515 = 0.97 (2 dp). c) Let T be the total number of defects in 50 products of size 4m by 10m. Then the mean is 40 x 0.3 = 12 and T is Po(50x12)= Po(600). Pr(T> 300) = 1 - P(T

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