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.

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,
$$
P(B,C|A) = 0.75\sqrt{0.8\times 0.2 \times 0.9\times 0.1} + 0.8\times 0.9 = 0.81.
$$
