Final answer:
The probability that a defective item is in a box given it is not found in the same box after inspecting n items is calculated by dividing the remaining unexamined items in the box by the total possible items it could be in. This calculation is the same across all boxes after checking the first box.
Step-by-step explanation:
Conditional Probability of Finding a Defective Item
Lets assume there are k boxes with m items in each box, making a total of km items. It is known that one item among these is defective. Suppose n items are randomly selected from each box for examination.
a) Probability that the defective item is in box 1, given it is not found in box 1
If the defective item is not found in box 1 after examining n items, it must be one of the remaining (m-n) items in box 1 or in one of the other (k-1) boxes. The total number of ways we could have not found the defective item in box 1 is:
(k-1)m + (m-n) = km - n
The probability of it being in box 1, given that it is not found there, is then the number of remaining unexamined items in box 1 divided by the total possible items it could be in:
P(Defective in Box 1 | Not found in Box 1) = (m-n) / (km - n)
b) Probability that the defective item is in box 2, given it is not found in box 1
The conditional probability that it is in box 2 is similar to box 1, except we focus on the m items in box 2. The probability remains the same since whether the box is labeled '1' or '2' doesn't change the outcome, so:
P(Defective in Box 2 | Not found in Box 1) = m / (km - n)