Hello, valosn!

Here's #1 . . .

1. A factory produces items in boxes of two.

Over the long run:

. . (A): 92% of the boxes contain 0 defective items,

. . (B): 5% of the boxes contain 1 defective item, and

. . (C): 3% of the boxes contain 2 defective items.

A box is picked at random from production, then an item is picked

at random from that box. Given that the item is defective, what is

the probability that the second item in the box is defective?

We are concerned with two events:

. . : the first item is defective, and

. . : the second item is defective.

We want: . .[1]

The numerator is: , the probability that both are defective ... case (C).

. . We are told that this happens: of the time.

Hence: .

The denominator is: , the probability that the first is defective.

This can happen in two ways:

(i) Case (B) which happens of the time

. . and we pick the defective item, probability .

Thus, this has probability:

(ii) Case (C) which happens of the time

. . and we pick a defective item, probability

Thus, this has probability:

Hence: .

Substitute into [1]: .