# Conditional probability calculation

This is a homework question about probability that I do not understand, even after discussing it with my teacher. I understand the correct answer, but don't understand why my answer is wrong.

A company buys rejected devices from two manufacturers. 60% of the devices are made by manufacturer A, 40% are made by manufacturer B. 10% of A's devices are defective. 30% of B's devices are defective.

A new shipment arrived. Due to sloppy handling, we don't know which device comes from which manufacturer.

We take a sample of 20 devices. They are tested thorougly. 2 devices are defective.

Question: What is the probability of finding two defects if we don't know who the manufacturer was?

My answer: general probability of finding a defect is $0.6\cdot0.1 + 0.4 \cdot 0.3 = 0.18$. Probability of finding two defects in a sample of 20 is then $\binom{20}{2} \cdot 0.18^2 \cdot 0.82^{18} = 0.1730$

This is not correct. The correct answer is to first find the probability of two defects when we know that the devices are made by A ($0.2852$) or B ($0.0278$). We then do $0.6\cdot 0.2852+0.4\cdot 0.0278=0.18224$.

I would like to understand why my method is wrong, what my error in thinking is.

Your method would be correct if (assumes that) each of the 20 devices in the sample could be from either manufacturer, independently of all the others in the sample, i.e., that the number of devices from manufacturer A is Binomial with parameters 20 and 0.6, with the remaining devices being from manufacturer B.

Based on the "correct" answer, it is apparently the case, although not explicitly stated, that all 20 devices in the sample are from the same manufacturer (A or B). I.e., with probability 0.6, all 20 are from manufacturer A, and with probability 0.4, all 20 are from manufacturer B.

• But the question actually states: "Due to sloppy handling, we don't know which device comes from which manufacturer". Why would a random subset of 20 samples all happen to be from the same manufacturer? – Moss Murderer Nov 1 '17 at 0:22
• That's why I wrote 'Based on the "correct" answer, it is apparently the case, although not explicitly stated ...' I.e., there seems to be an assumption that the entire shipment, form which 20 devices are selected, is from a single manufacturer, we just don't know which manufacturer. I grant you that it is a sloppily written question, requiring you to use the answer to determine what the complete problem statement (assumptions) must be. – Mark L. Stone Nov 1 '17 at 0:35
• You'll notice I put correct in quotes. – Mark L. Stone Nov 1 '17 at 0:41
• Yes I noticed it. My comment was not meant as a critique but rather to highlight that the wording of the question actually gives enough reason to assume the contrary. – Moss Murderer Nov 1 '17 at 0:44
• Aaaah! Thanks so much, this has been bugging me for days. English is not my native language and I translated the question. The original question is (in hindsight) clear about this assumption. I do wonder now if technically the answer is wrong because the conditional probabilities are calculated with the binomial formula, but this is a finite population and we explicitly learned that if n=10N n is not big enough to assume it doesn't matter (n should be > 10N). So, shouldn't P(2D|A) be 20*0.1^2*0.9^18+19*0.1*0.9^18=0.3152 instead of 0.2852? – broccoli Nov 1 '17 at 7:50