Calculating probabilities when events are neither independent nor mutually exclusive

Question

Consider three events: $E_1$, $E_2$ and $E_3$. They are neither mutually exclusive nor independent. Let's say that the correlation coefficient between $E_i$ and $E_j$ is given by $\rho_{ij}$. Let's also assume that the probability of $E_i$ occurring is $P(E_i)$.

How do we calculate the probability that at least one of these three events occurs? Any references that develop a general methodology to calculate probabilities when events share these two characteristics (mutually non-exclusive and correlated)?

Any help much appreciated! Thanks in advance.

Answer 1

The probability of at least one event occurring is found by the principle of inclusion and exclusion:

$$\mathsf P(E_1\cup E_2\cup E_3)=\sum_{i=1}^3\mathsf P(E_i)~-~\sum_{1\leqslant i<j\leqslant 3}\mathsf P(E_i,E_j)~+~\mathsf P(E_1,E_2,E_3)$$

But... You do not have enough information.

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If you know the marginal probabilities and correlation coefficients, then you can find the three pairwise joint probabilities: $\mathsf P(E_i,E_j)$.

$$\begin{align} \rho_{ij}&=\dfrac{\mathsf P(E_i,E_j)-\mathrm P(E_i)\mathsf P(E_j)}{\sqrt{(\mathsf P(E_i)-\mathsf P(E_i)^2)(\mathsf P(E_j)-\mathsf P(E_j)^2)~}} \\[3ex] \therefore\qquad\mathsf P(E_i,E_j)&=\mathrm P(E_i)\mathsf P(E_j)+\rho_{ij}\sqrt{(\mathsf P(E_i)-\mathsf P(E_i)^2)(\mathsf P(E_j)-\mathsf P(E_j)^2)~} \end{align}$$

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Unfortunately, this will not provide the tripplewise joint probability: $\mathsf P(E_1,E_2,E_3)$ and this is needed too.