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another probability question... thank you so much for helping... thanks a lot... 
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A manufacturer of electronic components sells 1KW resistors in boxes of 12. A customer requires the components to meet a particular specification, and is happy to accept boxes containing no more than 1 which fail to comply. When the manufacturer tests 1000 of his resistors he finds that 94 do not meet the specification. If he supplies 200 boxes to the customer, how many can he expect to be rejected?
A trainee glass-cutter has the job of cutting up large sheets of glass to make small panes of specific sizes. On average, for every 10 panes he cuts correctly he breaks one. He has a sheet large enough to produce 10 panes, from which he needs to get 9 correctly cut. What is the probability that he achieves this, assuming that the cuts are statistically independent?
1) By the test, the chances of the resistor failing is .094. Looking at one box of 12, there are three different cases we should look at:
a) all 12 things are fine
b) one of the 12 is bad, but the other 11 are fine
c) more than one is bad and the box gets rejected.
It's pretty challenging to calculate (c) directly because it has a lot of cases (2 bad, 3 bad, 4 bad, ...), so instead, let's calculate (a) and (b) and subtract.
a) The probability that one thing is good is 1-.094 = .906. So the chances that they are all good is .906^12 = .306.
b) The probability that exactly one is bad follows a binomial distribution, which is the same as a 12 independent trials of a bernoulli random variable. Here our probability is
(.906)^11*(.094)*(12 choose 1) = .381. What this is really saying is 11 are good with prob. .906, 1 is bad with prob. .094, and the (12 choose 1) picks which one is the bad one.
c) That means the probability that the box is rejected is 1 - .306 - .381 = .313, so of the 200 boxes, we'd expect 200*.313 to get rejected, which is about 63.
2) This is a very similar type of problem. In every 11 panes, 10 are good but one is bad, so his probability of cutting it successfully is ~.9091. (The question is worded a bit oddly--I think that's what it means, not that he breaks one out of every 10. If you think that's what it is, then use .9 instead.) Now he's going to do this 10 times, so either:
a) he gets them all right
b) he breaks 1
c) he fails.
a) is .9091^10 = .3856
b) is (.9091)^9*(.0909)*(10 choose 1) = .3856
This time we want the chances that he succeeds, so its' .3856+.3856 = .7712.
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