In a manufacturing process for electrical components, the probability of a finished component being defective is 0.012, independently of all others. Finished components are packed in boxes of 100. A box is acceptable if it contains not more than 1 defective component. Justify the use of a Poisson approximation for the distribution of the number of defective components in a box. 8 boxes are randomly chosen. Given that all of them are acceptable, estimate the conditional probability that they contain exactly 6 defective components altogether.
The Attempt at a Solution
Let X be the number of defective components in a box containing 100 components, then X~B(100,0.012) => X~P(1.2)
P(X<=1) = 0.663
Let Y be the number of acceptable boxes in 8 randomly chosen ones, thus Y~B(8,0.663)
P(Y=8) = 0.663^8 = 0.0372
P(required) = P(8 acceptable boxes contain 6 defective components) / P(8 acceptable boxes)
I could calculate the probability that 8 randomly chosen boxes contain 6 defective components (using a Poisson distribution with mean=9.6, I guess), but I'm clueless when it comes to '8 acceptable boxes'.
Any feedback would be highly appreciated. Thanks!