Relationship between conditional probability and correlation

Question

Is there a relationship between the conditional probability between two events and the correlation between them? For example, is it more likely there is a positive or negative correlation if the conditional probability reaches a certain point?

Answer 1

By the correlation between two events $A$ and $B$, I presume you mean the Pearson correlation coefficient between their indicator random variables $I_A$ and $I_B$. We have

$$ \text{Cov}(I_A, I_B) = \mathbb E[I_A I_B] - \mathbb E[I_A] \mathbb E[I_B] = \mathbb P(A \cap B) - \mathbb P(A) \mathbb P(B) = \mathbb P(A) \mathbb P(A^c) (\mathbb P(B \mid A) - \mathbb P(B \mid A^c)) $$ $$ \sigma(I_A) = \sqrt{\mathbb P(A) \mathbb P(A^c)}$$ and similarly for $\sigma(I_B)$, so the Pearson correlation coefficient is $$ r = \sqrt{\frac{\mathbb P(A) \mathbb P(A^c)}{\mathbb P(B) \mathbb P(B^c)}} (\mathbb P(B\mid A) - \mathbb P(B \mid A^c))$$

In particular, $r > 0$ iff $0 < \mathbb P(A) < 1$ and $\mathbb P(B \mid A) > \mathbb P(B \mid A^c)$.