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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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  • $\begingroup$ Correlation is a pairwise property. Are you wanting to assume that all three correlations have the same (positive) value, or just that all three have positive value? $\endgroup$ – Dilip Sarwate Mar 25 '15 at 13:21
  • $\begingroup$ Either, really. For simplicity's sake, let's assume that there is a correlation of 0.1 between A and B, 0.1 between B and C and 0.1 between A and C. $\endgroup$ – Hugh Mar 25 '15 at 13:26
  • $\begingroup$ Bonferroni's Inequality has few assumptions and gives a bound that is sometimes useful (depending on the size of the probabilities). Otherwise, I think you need to provide some context for your question. As stated, I do not see how a useful answer can be given. $\endgroup$ – BruceET Mar 25 '15 at 16:21
  • $\begingroup$ I just had the same question and came over this here. The question is about extending the problem of finding the probability of two events happening both (while knowing the probabilities of them happening and correlation) to three events. I.e. you have the probabilities of all three events and their correlation matrix and you would like to know the probability of all three occuring. $\endgroup$ – josefec Jul 23 '15 at 16:44
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    $\begingroup$ There are six parameters here $P(A), P(B), P(C), \rho_{AB}, \rho_{BC}, \rho_{AC}$, and eight unknown probabilities $P(A \cap B \cap C), P(A \cap B \cap \overline{C})$, etc. The six parameters give six constraints on the probabilities, and the fact that the probabilities need to sum to one gives a seventh. Since we have seven equations in eight unknowns we should in general expect a one-parameter family of solutions. $\endgroup$ – Michael Lugo Aug 7 '18 at 20:10
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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.

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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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