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We incorporate statistical confidence intervals in both the multi-armed bandit and the reinforcement learning problems. In the bandit problem we show that given n arms, it suffices to pull the arms a total of O ((n/ε2) log (1/δ)) times to find an ε-optimal arm with probability of at least 1− δ. This bound matches the lower bound of Mannor and Tsitsiklis (2004) up to constants. We also devise action elimination procedures in reinforcement learning algorithms. We describe a framework that is based on learning the confidence interval around the value function or the Q-function and eliminating actions that are not optimal (with high probability). We provide a model-based and a model-free variants of the elimination method. We further derive stopping conditions guaranteeing that the learned policy is approximately optimal with high probability. Simulations demonstrate a considerable speedup and added robustness over ε-greedy Q-learning.