# Calculating probabilities when events are neither independent nor mutually exclusive

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.

• There is not enough information to find this probability. Aug 30 at 7:39
• Perhaps $E_1$, $E_2$ and $E_3$ take values in $\{0,1\}$, and the probability of $E_i$ occurring is $P(E_i=1)$. That seems to make sense. Aug 30 at 8:00

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

But... You do not have enough information.

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}

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

• Thank you! I think knowing a few conditional probabilities between those events takes us a step closer, since we can then use this: math.stackexchange.com/questions/1281454/… Aug 30 at 10:30