Let X1, X2, . . . be i.i.d. random variables such that $P(X_1 = -1) = 2/3$ and $P(X_1 = 2) =1/3$
and for each integer $n \geq 1$ put $S_n := X_1 +X_2 +\cdots+X_n$. For a sufficiently large $n$, which of the following events is more likely $\{S_n \geq n/2\}$ or $\{S_n \leq -n/2\}$?
My Attempt: Use large deviation and see that the probability for the first event is $e^{-nI(n/2)}$ and the probability for the second is $1-e^{-nI(-n/2)}$. I know that both decrease to 0 but with different rates. But how do I tell which one decreases faster?