You repeatedly evaluate a certain discrete R.V. (e.g., discrete triangular) $X_i$ (i.i.d. from one another). What's the probability that two consecutive $a$'s appear before a $b$ appears ($a$ and $b$ are particular realizations that $X_i$ can take).
If it were immediately before a $b$, I assume it would be simply $P(X_i=a)^2$, but I am not restricting it to strictly immediately before $b$.
Would it be correct to use the binomial distribution with $$n= 1/P(X_i = b) - 2$$ The geometric mean. Subtract 2 because I ignore the slot when $b$ occurs and am looking at two consecutive slots to fill, so the first slot doesn't count) and
$$p = P(X_i=a)^2$$ ?
I've seen a much simpler way for this type of question that appears to use some sort of version of the law of total probability (including a recursion term, i.e., the probability being asked for itself), but I wasn't able to see the rationale behind it.