 Solution

Let $Z$ be a fair Bernoulli variable measurable at time zero and let $(S_n)$ be an independent simple symmetric random walk. Then $X_n=Z{S}_n$ is a martingale with increments bounded by one. On $\{Z=0\}$ it converges to zero, while on $\{Z=1\}$ the recurrence of the simple symmetric random walk gives limsup $+\infty$ and liminf $-\infty$. Thus $\mathbb{P}(A)=\mathbb{P}(B)=1/2$.