I am facing a problem that I cannot find the answer to. I have three variables, A, B and C. There are only two possibilities for each of these, A either happens or it does not, B happens or it does not and C happens or it does not. I know that if these events are independent that the probability of them all occurring is simply $P(A)\cdot P(B)\cdot P(C)$. So if the probability of each happening is 10% then all three have a $10\%·10\%·10\% = 0.1\%$ probability of occurring. But how would this formula change if the events were not independent but were instead positively correlated.
I can solve this for just two variables with the formula: $P(A \cap B) = P(A)\cdot P(B) + \rho_{AB}\cdot \sqrt{P(A)\cdot (1-P(A))\cdot P(B)\cdot (1-P(B))} $, where $\rho_{AB}$ is the correlation coefficient between A and B.
How would I change this formula to calculate the probability that A, B and C all occur? I.e. calculating $P(A \cap B \cap C)$ knowing $P(A)$, $P(B)$, $P(C)$, $\rho_{AB}$, $\rho_{AC}$, $\rho_{BC}$.
Thanks in advance for the help!
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$\begingroup$As I found out in this presentation, where the above-mentioned issue is dealt with in the context of default correlations, the problem is not possible to be solved with the inputs given.
I.e., when 3 events have defined probabilities of happening, the pairwise correlations are not enough to construct the joint probability of all 3 events happening.
In the context of default correlations, complex problems of correlations of events are solved using other methods such as Copula functions.
$\endgroup$ $\begingroup$I believe you can take the average correlation and plug it in while adding the addition P(C) & P(C)⋅(1−P(C)) to each side of the equation.
This at least works for my use case.
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