Worthless and Weak

Lets say that you have a hell of a lot of boxes. A million. Or a billion. And inside each of these boxes is either a substance, we'll call it X, or a substance that we'll call Y.

Now, you can't open the boxes, but you do have an 'X' detector. It will give you a one of two responses. Either it will say "this box contains X" or "I do not know whether the box contains X."

The first go around, as you sample boxes, you get about a .2% rate of "X" and a 99.8% rate of "unknown."

The days and the years go by, and they build better and better X detectors (ones with a lower false negative rate). You begin to see higher and higher rates of "X" 1% or 4%, or eventually 10%.

At which success level is it reasonable to assume that every box contains X?

Lets assume a 0% failure rate. (Or rather, what this problem really is a 0% false positive rate, with an unknown false negative rate).

I don't really have any idea to go about this problem. Obviously, once it hits 100%, we can assume X. Other than that, I would love to hear ideas if you have any.

Labels: math problems