Does selecting the same arm has the same reward?
In the version of the Multi-Armed Bandit problem I'm familiar with, there is a fixed list of distributions $B = R_1, R_2 \cdots R_n$, and the reward for pulling lever $k$ is chosen from the distribution $R_k$.

So pulling the same lever twice might lead to different rewards, but the rewards are connected through the distribution $R_k$.