# Probability that the third student didn't answer correctly given that only 2 out of 3 students answered correctly

I tried solving this but failed many times. I couldn't really wrap my head around the right point to start. Here's the question:

Students A, B, and C each independently answer a question on a test. The probability of getting the correct answer is 0.9 for A, 0.7 for B, and 0.4 for C. If 2 of them get the correct answer, what is the probability C was the one with the wrong answer?

Even hints would suffice. Not really necessarily looking for the answer but just the correct headstart. Thanks.

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This is perhaps not the most elegant approach, but if you’re having trouble with this type of problem, it may help. There are three possibilities: A had the wrong answer, B had the wrong answer, and C had the wrong answer. Calculate the probabilities of these three events; call them $p_A,p_B$, and $p_C$, respectively. The probability that you’re in one of these three situations is $p_A+p_B+p_C$. What fraction of that probability is due to C’s being the unlucky student? It’s

$$\frac{p_C}{p_A+p_B+p_C}\;.$$

That’s the probability that you want.

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Yay! Thanks a lot! Got the correct answer! – Potato Queue Feb 4 '13 at 1:19
@PotatoQueue: Excellent! You’re welcome. – Brian M. Scott Feb 4 '13 at 1:19