I have a problem in maths.

Suppose we have two magicians, Micky and Macky.

Micky tells the truth, when asked a question, with probability x and Macky tells the truth with probability y.

When Micky and Macky are asked about the state of a certain electrical switch they are both affirmative and they say state "A" is the state at which our proverbial switch is right now.

What is then the probability p(A) ?

A simple answer is they are either both telling the truth or both lying.

So:

P(A) = x.y / [ x.y + (1-x).(1-y) ]

But what happens if there is some kind of unknown correlation between the two of them ?

Macky is sometimes influenced by Micky's decision to tell the truth or to lie, in a certain peculiar way.

We can observe the final outcome i.e. measure p(A) but we don't know the inner reasoning.

Or perhaps we can work it out but what if we have many magicians m1, m2, m3 ... and cannot hope to make as many measurements as may be needed ?

Can someone help me with this ?

I am trying to google out some answer but I don' t seem to be able to put the right catch phrases.

p.s. perhaps I will think of some more representative example of the statitical proble I 'm trying to solve and come back, if the above does n't help.