The fifth item inspected

Question

A tray has 12 items, of which 3 are defective and 9 are good. An inspector checks them one at a time in a random order. What is the probability that the fifth item inspected is defective?

Accepted Answer

You could try to reason about the first four inspections, but symmetry makes that unnecessary. Line the 12 items up in a random order. Every arrangement is equally likely, so the fifth slot is just as likely to hold any particular item as any other slot. That makes the fifth item a fair random pick from all 12. So the chance it is defective is the overall defective fraction: P(	ext{5th is defective})=\frac{3}{12}=\frac{1}{4}=0.25. Takeaway. The "which position" detail is a distraction. Any fixed position is a uniformly random item, so its chance of being a given type is that type's share of …

The fifth item inspected

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