# Relationship between conditional probability and correlation

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?

• What's "the conditional probability between two events"? Commented Feb 2, 2020 at 23:19
• P(A | B) = the probability of A, given B Commented Feb 2, 2020 at 23:45
• The correlation between $A$ and $B$ is symmetric in $A$ and $B$, whereas $P(B\mid A)\ne P(A\mid B)$, so it makes little sense to call one of the two "the conditional probability between two events". What would you call the other one then? Commented Feb 2, 2020 at 23:48

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 $$\mathbb P(B \mid A) > \mathbb P(B \mid A^c)$$.
• No, it is not. $P(B \mid A)$ could be as small as you want, as long as it's greater than $P(B \mid A^c)$. Commented Feb 4, 2020 at 2:58