Event A Before Event B

Let $(X_n)_{n \geq 1}$ be a sequence of i.i.d. random variables taking values in the set $\{A, B, C\}$ , with:

\mathbb{P}(X_1 = A) = p_A, \quad \mathbb{P}(X_1 = B) = p_B, \quad \mathbb{P}(X_1 = C) = p_C

where $p_A, p_B, p_C \in [0,1]$ and (with and ).

The experiment is repeated until either $A$ or $B$ is observed. Calculate the probability that $A$ appears before $B$ .

Answer