Probability Q - The vials issue

[snomer]

Hi,

This is actually a problem I'm dealing in work.

we manufacture about 1000 vials per batch. 100 of them are rejects. however we managed to rationalize the use of rejects in quality control (QC) tests.
the QC tests requires 25 vials out of each batch. So by using 25 rejects, we save 25 vials (which might not sound a lot, but it is quite a lot of money).
However, the problem is that we must sample 4 vials from the beginning of the filling event, 4 from the middle and 8 from the end of the filling event.

That means we have to sample 16 specific vials, which not always are rejects. In order to increase the chances that more rejects are picked, we want to widen the sampling range. For example: instead of sampling first 4 vials, sampling of 4 vials out of the first 100 vials will be allowed. Needless to say there are restrictions for widening the range so we can't just decide to eliminate the requirement.

All in all, my question is, how do I even calculate the probability to pick 4 vials out first 100 vials, 4 vials of mid 100 vials and 8 out of last 100 vials ,while p is the probability for picking a reject. Since we sample, there is no replacement (the p is not constant).

Thanks,
Snomer

 

[Steven G]

I am not sure that I am understanding what you are saying. You have 10% of the vials being rejects (100 out of 1000 is 10%). It doesn't matter how you pick vials for testing, 10% will be rejects.

It like you toss 100 coins and write down for 10 of the tosses whether it was heads or tails. 50% will be heads!

 

[Dr.Peterson]

we manufacture about 1000 vials per batch. 100 of them are rejects. however we managed to rationalize the use of rejects in quality control (QC) tests.
the QC tests requires 25 vials out of each batch. So by using 25 rejects, we save 25 vials (which might not sound a lot, but it is quite a lot of money).
However, the problem is that we must sample 4 vials from the beginning of the filling event, 4 from the middle and 8 from the end of the filling event.

That means we have to sample 16 specific vials, which not always are rejects. In order to increase the chances that more rejects are picked, we want to widen the sampling range. For example: instead of sampling first 4 vials, sampling of 4 vials out of the first 100 vials will be allowed. Needless to say there are restrictions for widening the range so we can't just decide to eliminate the requirement.

All in all, my question is, how do I even calculate the probability to pick 4 vials out first 100 vials, 4 vials of mid 100 vials and 8 out of last 100 vials ,while p is the probability for picking a reject. Since we sample, there is no replacement (the p is not constant).

I'm trying to figure out what probability you are asking for, and how it affects what you are doing.

First, I assume the "rationalization" is that somehow the criterion on which vials are rejected is independent of what QC is testing for; but you still don't want to randomly choose 25 of the 100 or so vials that have already been rejected. So you are doing a systematic sample from everything, and hoping that as many as possible will turn out to be rejects. Is that right?

It seems to me that whether you sample vials 1, 2, 3, and 4, or vials 6, 17, 42, and 89, you will have the same distribution of rejects; that is, this won't change the probability of at least one being a reject, or the expected number of rejects, or anything like that. This is random, after all.

So the answer may well be that nothing you do can increase the number of rejects you will choose randomly. But maybe I don't understand the situation.