Let X and Y be Bernoulli random variables. We don't assume independence or identical distribution, but we do assume that all 4 of the following probabilities are nonzero.
Let a := P[X = 1, Y = 1], b := P[X = 1, Y = 0], c := P[X = 0, Y = 1], and d := P[X = 0, Y = 0].
How do I obtain a formula for a correlation between random variables X and Y?
On
Presumably you're talking about the Pearson correlation coefficient. It's defined in terms of the covariance and the standard deviations. These in turn are defined in terms of expected values, which are defined in terms of probabilities. For example, $E[X] = 1 \cdot P(X=1) + 0 \cdot P(X=0) = a + b$.
Hint: The correlation is defined in terms of $\mathrm{Cov}(X,Y)$, $\mathrm{Var}(X)$ and $\mathrm{Var}(Y)$ which can be computed if we know the following quantities $$ {\rm E}[X],{\rm E}[X^2],{\rm E}[Y],{\rm E}[Y^2],{\rm E}[XY]. $$ These should be straightforward to find. For instance, $$ {\rm E}[X^2]={\rm E}[X]=P(X=1)=a+b. $$