Joint probability of two paired conditional probabilities

Suppose that the probabilities of the positive outcomes of B and C condition on the positive outcome of event A are:

$$p_1 = Pr(B+|A+) = 0.8$$

$$p_2 = Pr(C+|A+) = 0.9$$

And we also know that the correlation coefficient between $$p_1$$ and $$p_2$$ is 0.75 (i.e. B and C are paired binary outcomes, e.g. a same cohort of patients tested for COVID-19 at day0 and day3). My question is what the joint probability of $$Pr(C+,B+|A+)$$ is in a closed form and the value in this case.

• That's insufficient information unless $\Pr(A+)=1.$
– whuber
Commented Aug 22, 2022 at 16:29
• Thanks for your comment. Let assume $Pr(A+) = 0.3$. Commented Aug 22, 2022 at 19:09
• That doesn't help. You need information equivalent to the conditional correlation to obtain a unique answer.
– whuber
Commented Aug 22, 2022 at 21:10

Let $$B|A\sim Bernoulli(p_B)$$ and $$C|A\sim Bernoulli(p_C)$$, and let $$Cor(B,C|A) = \rho.$$ We know that \begin{aligned} Cor(B,C|A) &= \frac{Cov(B,C|A)}{\sqrt{Var(B|A)Var(C|A)}}\\ &=\frac{E[BC|A]-E[B|A]E[C|A]}{\sqrt{Var(B|A)Var(C|A)}}\\ &=\frac{E[BC|A]-p_Bp_C}{\sqrt{p_B(1-p_B)p_C(1-p_C)}}\\ \end{aligned} Note that $$E[BC|A] = P(B = 1, C = 1|A)$$, for in any other case, the product $$BC$$ is equal to zero. This is, if $$C = 0$$ or $$B = 0$$ then $$BC = 0$$. Only when both are one we have that $$BC = 1$$. Thus, using the above and solving for $$E[BC|A]$$ we have that \begin{aligned} P(B = 1, C = 1|A) &= E[BC|A]\\ &= \rho \sqrt{p_B(1-p_B)p_C(1-p_C)} + p_Bp_C. \end{aligned} Given the data, $$\rho = 0.75$$, $$p_B = 0.8$$ and $$p_C = 0.9$$. Thus, $$P(B,C|A) = 0.75\sqrt{0.8\times 0.2 \times 0.9\times 0.1} + 0.8\times 0.9 = 0.81.$$
• It states that the correlation between $p_1$ and $p_2$, the conditional probabilities, is 0.75, so I assumed the author refers to the conditional correlation. But indeed, if it is not, then we do not have enough information. Commented Aug 22, 2022 at 21:56