mei stats 1 question trouble

A particular product is made from human blood given by donors. The product is stored in bags. The production process is such that, on average 5% of the bags are faulty. Each bag is carefully tested before use.

A random sample of n bags is selected. The production manager wishes there to be a probability of one third or less of finding any faulty bags in the sample. Find the maximum possible value of n, showing you working clearly.

Me and my teacher spent about half an hour on this question and neither of us could do it. We have the answers, but we dont understand what is going on:

1-0.95^n =< 1/3
0.95^n >= 2/3
n =< log2/3 /log0.95, so n =< 7.90

maximum is n=7

Basically, at the start of the working we think that the 1-0.95^n should be 0.05^n and we dont understand why it is not.

Thanks for any help you can give :smile:

Daniel

Reply 1

generalebriety

If you want the probability of finding a faulty bag to be less than a third, you want the probability of finding no faulty bags (=0.95^n) to be greater than two thirds. That's why their answer's right.

You were doing 0.05^n <= 1/3. Think about what that means - you want the thing that happens with probability 0.05 to happen n times with probability less than a third. In other words, you want a probability of less than a third that all the bags are faulty, i.e. a probability of more than two thirds that there's at least one non-faulty bag. (Edit: anyway, work your calculation through. You'll get n as some decimal much less than one, which, while not really an answer to your question, should be enough to convince you that you've done something silly.)

Think binomial distribution (or, if you can't, tell your teacher to).