Definition of correlation between events? I often hear people talk about correlation between events, e.g. event A and event B are positively correlated.
However, unlike correlation between random variables (i.e. $\frac{Cov(X,Y)}{\sigma_X \sigma_Y}$ ), I can't seem to find a numerical definition of correlation between events.
In this book, it is mentioned that

*

*$\mathbb{P}(∣)$ > $\mathbb{P}()$ if and only if $\mathbb{P}(∣)$ > $\mathbb{P}()$ if and only if $\mathbb{P}(∩) > \mathbb{P}()\mathbb{P}()$. In this case,  and  are positively correlated.

*$\mathbb{P}(∣) < \mathbb{P}()$ if and only if $\mathbb{P}(∣) < \mathbb{P}()$ if and only if $\mathbb{P}(∩) < \mathbb{P}()\mathbb{P}()$. In this case,  and  are negatively correlated.

*$\mathbb{P}(∣) = \mathbb{P}()$ if and only if $\mathbb{P}(∣) = \mathbb{P}()$ if and only if $\mathbb{P}(∩) = \mathbb{P}()\mathbb{P}()$. In this case,  and  are uncorrelated or independent.

However, there is still no numerical definition of correlation between events.
What is the numerical formula definition of correlation between events? Is it a number between -1 and 1?
 A: It seems that the book gives a definition of correlation between two events that amounts to saying that two events are correlated if they are not independent.
The quantity $\frac{\text{Cov}(X,Y)}{\sigma_X\sigma_Y}$ is the Pearson Correlation Coefficient and serves as a numerical measure of linear correlation between two random variables. There are other kinds of correlation coefficients that might be useful in other applications, though I am not as familiar.
I suppose if you wanted a numerical measure of the linear correlation between two events, $A,B$, you could consider the pearson correlation coefficient of the r.v. indicators of these events, $I_A$ and $I_B$. Using $\text{Cov}(I_A,I_B) = \text{E}[I_AI_B] - \text{E}[I_A][I_B]$ and the fact that $\text{E}[I_A] = \mathbb{P}(A)$, $\text{E}[I_B]= \mathbb{P}(B)$,  and $\text{E}[I_AI_B] = \mathbb{P}(A\cap B)$ one arrives at $\text{Cov}(I_A,I_B) = \mathbb{P}(A\cap B) - \mathbb{P}(A)\mathbb{P}(B)$. Using $\sigma_{I_A}^2 = \mathbb{P}(A)(1-\mathbb{P}(A))$ and a similar formula for $\sigma_{I_B}^2$, you can calculate the pearson correlation coefficient.
