Let $X_1, X_2, ...$ be iid random variables with $$P(X_j = 2)= 1/3, P(X_j = 1/2) =2/3.$$ Let $M_0 = 1$ and for $n \geq 1, M_n = X_1X_2 \cdots X_n$.
Let T be the first time that $M_n = 4$. Is it true that $P(T<\infty)=1$? How to show that? My ultimate goal is to find the probability that $M_n$ ever reaches $4$ by optional stopping theorem.