Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed with $P\left(X_i=1\right)=P\left(X_i=-1\right)=p$ and $P\left(X_i=0\right)=1-2 p$ for all $i=1,2, \ldots, n$. Define
$$
a_n=P\left(\prod_{i=1}^n X_i=1\right), b_n=P\left(\prod_{i=1}^n X_i=-1\right) \text { and } c_n=P\left(\prod_{i=1}^n X_i=0\right) \text {. }
$$
Which of the following is true as $n$ tends to infinity?
(A) $a_n \rightarrow 1 / 3, b_n \rightarrow 1 / 3, c_n \rightarrow 1 / 3$
(B) $a_n \rightarrow p, b_n \rightarrow p, c_n \rightarrow 1-2 p$