Correlation in errors

I'm not good in statistics, so please excuse my noob question. We want to ask a question from people (say what is $2+2$). They might make mistake. We assume that they give the correct answer with the same probability $p$. In case their error rates are not correlated, the probability that they both give the wrong answer is $(1-p)^2$ However, their error rates are not independent, they are correlated. In this case, how can we reason about the accuracy of the result ? Do we need to know the distribution of answers?

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2 Answers

Well, if the errors can be correlated, it is possible that both make mistake at the same times, in which case the probability that both give the wrong answer is $1-p$. On the other hand, the minimum probability that both are wrong happens when the sets of cases where the two give right answers are as disjoint as possible, in which case the probability is $\min (0, 1-2p)$.

So really all we can say about this probability without any further knowledge about correlation of error rates is that it must lie between $\min (0, 1-2p)$ and $1-p$ (both inclusive).

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Your answer isn't clear. Mine is much more specific aby providing the solution as a function of the unknown conditional probabilities. – Michael Chernick Jul 6 '12 at 20:20

What you need to know are the the following conditional probabilities

1. $p_1=P(A \text{ is correct} |B \text{ is correct})$
2. p2=P(B is correct| A is correct)
3. p3=P(A is incorrect|B is correct)
4. p4=P(B is incorrect |A is correct) Since we know P(A is correct) =P(B is correct)=p the other conditional probabilities can be obtained using relationships like p=P(A correct)= p1 p + P(A correct| B is incorrect) (1-p).

These conditional probabilities determine the degree to which A and Bs responses are correlated. The probability that both give the wrong answer P(A is incorrect) P(B is incorrect|A is incorrect) =(1-p)(1- {p p4/(1-p)}) =1-p(1+p4)

This results because P(Bis incorrect)=1-p= P(B is incorrect|A is correct)p +(1-p) P(B is incorrect|A is incorrect)= p4 p+(1-p) P(B is incorrect|A is incorrect) and we get P(B is incorrect| A is in correct)=1 - p p4/(1-p).

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For the record, I didn't downvote you. – narengi Jul 23 '12 at 16:07
@narengi Did you like the answer? – Michael Chernick Aug 15 '12 at 21:00