I am trying to calculate the variance of repeated, independent sampling of two mutually exclusive events, $X$ and $Y$. For example, $X$ could represent rolling a fair $6$-sided die and $Y$ rolling a loaded die that only lands on $2,3,4,5 or 6$ (all with $20%$ probability). If each time you roll, you reach into a bag and select either $X$ or $Y$, what is the variance of $n$ rolls?
For the example given and others like it, I know how to manually calculate the variance by calculating the expected value, the probability of each outcome, and the deviation of each outcome from the mean. However, for more complex problems this quickly becomes very tedious.
So, given $P(X) = p$ and $P(Y) = 1-p, Var(X), Var(Y)$, and any other necessary properties of these distributions, is there a formula to calculate the variance of n events without referring to the probabilities of individual outcomes (i.e. rolling a $3$)?