I was thinking about this problem that popped into my mind. Let $X_i$ and $Y_i$, $i = 1,...,n$ be two realizations of different random variables. Let's drop the i.i.d. assumption, for this case. Each realization of the two different random variables has a binary label, i.e. $L_{X_i}, L_{Y_i} \in \{0,1\}$.
Now, let $\rho = \text{Corr}(X_i,Y_i)$ denote the correlation coefficient between the two random variables.
If I know that I can correctly classify $X_i$ with a probability $p$, is it possible to compute analytically the probability of correctly classifying $Y_i$?
I don't even have an idea about how to attempt a solution to this problem to be honest, and I hope that somebody can help :)
Not without more info about the relation between the value of the random variable and their labels, you can't! Let's consider two extreme cases. In both cases I assume that $X_i$ and $Y_i$ take values in $\mathbb{R}$. Let's assume moreover that correlation is perfect, or even stronger that $X_i = Y_i$ for every $i$.
Case 1: labels are assigned to the $X_i$ independent of the value of the $X_i$ and of the labels of the other $X_j$ in such a way that $Prob(L_{X_i} = 0) = p$. Similarly each $Y_i$ is assigned a label independent of its value with $Prob(L_{Y_i} = 0) = q$. Your classification algorithm is to ignore the actual value of $X_i$ and $Y_i$ and just predict that the label is always $0$. Then you correctly classify the $X_i$ with probability $p$ but there is no way you could compute $q$ from this info.
Case 2: There is some unknown threshold $T$ such that $X_i$ get's assigned label $1$ if $X_i > T$ and $X_i$ gets assigned label $0$ if $X_i \leq T$. Labels of $Y_i$ are assigned in exactly the same way. You don't know $T$ and 'guess' that the label will be $0$ if $X_i \leq S$ for some other threshold $S$ and that the label will be $1$ if $X_i > S$ and you do the same for the $Y_i$. Now when $S \neq T$ there will be some mistakes meaning that $p < 1$. Suppose that some all-knowing being now tells you the value of $p$. Then you still don't know $T$ or anything else that might be useful in practice, but at least you now know that the probability of correctly classifying the $Y_i$ with this method, as that also equals $p$ in this case.