I have 3 boolean variables X,Y,Z (indicating if the same event has happened).
Let's assume the error of each of them is 0.3. This means, for example, that if X is true then the probability of an event is 70%. When X is false, the probability of an event is 30%. Same goes for Y,Z.
I have to find the probability(P) that the event has happened, given X,Y,Z have all indicated true.
If X,Y,Z were all independent (Pearson = 0) - then P = 1 - 0.3*0.3*0.3 = 0.973.
If X,Y,Z were all fully dependent (Pearson = 1) - then P = 1 - 0.3 = 0.7.
My question is: what is P, given the "Pearson correlation" between each pair((X,Y)(X,Z)(Y,Z)) is 0.5?