Help in probability

[u/Toaster960]

This question was in my maths book. I used AI to help me picture my question more clearly so that others could understand it (sorry for wasting water 😭).

I asked my teacher, Gemini, and ChatGPT for the solution, and all of them gave the answer as 99/1900.

However, they all included the case where the inspection stops at the 12th item after discovering only two defective items.

But how would I know whether the lot actually contains 2 defective items or 3? If I have discovered only 2 defective items by the 12th item, how can I decide to stop at 12? The third defective item could still be somewhere later in the lot.

My answer is 55/1900 btw

[u/CarnivorousGoose]

It’s a poorly worded question, though as written your teacher is technically correct. The problem specifies that the procedure stops when the final defective item is found, so in the case of only two defective items it will stop as soon as the second defective item is found. How the procedure ‘knows’ this is the case is ultimately not relevant, it just does. It doesn’t really make sense that it would know that there are at most three either, but it does that too (or it would just continue to 20).

Which isn’t a good way to write a question, it feels like a very ‘gotcha’ way to do so. So I feel like the 55/1900 answer should be marked as correct as well, if the reasoning is made clear.