Can someone help me on this probability question please?

Question

I'm quite comfortable with probability, but sometimes the wording of the questions REALLY throw me off. Given the following problem:

On a production line, $12\%$ of items are imperfect, and $25\%$ of these are rejected. Perfect items are never rejected. If $3$ items are selected at random, find the following probabilities:

i. The first item is rejected.

ii. No item is rejected.

The part I'm stuck at is the "$25\%$ are rejected" part. Is it:

1. $P(I\cap R) = 0.25$ or
2. $P(R\mid I) = 0.25$

Where $R$ is rejected and $I$ is imperfect.

I'm not asking for a solution to the problem. I just need help with the wording of the question.

Answer 1

25% of the imperfect items are rejected. 25% of overall items are rejected makes no sense since it clearly states that perfect items are never rejected and you would necessarily reject some perfect items otherwise. $P(R|I)=0.25$

Remember that $P(A\cap B)\leq P(A)$

$0.25=P(I\cap R)\leq P(I)=0.12$ yields a contradiction.

Answer 2

The exercise says: 12% of items are imperfect, and 25% of these are rejected. Rephrasing: given that an item is imperfect, there is $25\%$ chance that it will be rejected. So, you are given that

1. $P(I)=0.12$ and
2. $P(R\mid I)=0.25$

From these you can find that $P(R\cap I)=P(I)P(R\mid I)=0.12\cdot0.25=0.03$.

Answer 3

An item can vary in two dimensions: perfect/imperfect and rejected/kept. You can draw out a 2D grid to model this:

The top-left box represents items that are perfect and kept. The number in the bottom-right indicates items that are imperfect and rejected. Top-right items are false-positives and bottom-left items are false-negatives. I include row sums to the right and column sums at the bottom

Here's how to fill this in:

Let's assume we have a population of 100 items. We're told 12% are imperfect, that's 12 imperfect items and 88 perfect items. These are row sums. These numbers exist irrespective of kept/rejected status.

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Next we learn that 25% of the imperfect items are rejected. This effects only the 12 items that are imperfect so we can ignore the 88 perfect items. 25% of the 12 items is 3. The remaining 9 items are kept:

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Finally, we we're told that no perfect items are rejected. This is easy:

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Now we know everything we need to answer the questions. For our hypothetical population of 100 items, 3 are rejected. In general, unless we know more about an item, we can say that it has a 3% chance of being rejected. From here you should have an easier time answering your two questions.

I personally like this approach because you don't need to memorize a formula to get this far.