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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?

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  • $\begingroup$ What's "the conditional probability between two events"? $\endgroup$
    – joriki
    Feb 2, 2020 at 23:19
  • $\begingroup$ P(A | B) = the probability of A, given B $\endgroup$
    – bambi
    Feb 2, 2020 at 23:45
  • $\begingroup$ 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? $\endgroup$
    – joriki
    Feb 2, 2020 at 23:48

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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 $\mathbb P(B \mid A) > \mathbb P(B \mid A^c)$.

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  • $\begingroup$ So, is it correct then to say that P(B | A) must be greater than 0.5 in order for r > 0? $\endgroup$
    – bambi
    Feb 4, 2020 at 1:16
  • $\begingroup$ No, it is not. $P(B \mid A)$ could be as small as you want, as long as it's greater than $P(B \mid A^c)$. $\endgroup$ Feb 4, 2020 at 2:58

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