Information theorists may already be familiar with this, but it's an interesting puzzle for non-information-theorists. Here are two questions:
First an easy one: Suppose you roll a coin 1000 times. What is the conditional probability that the first roll is heads given that you observe 700 heads out of the 1000 rolls?
Next a hard one: Suppose you roll a die (faces numbered 1 through 6) 1000 times. What is the conditional probability that the first die roll lands on a 2 given that the average of 1000 rolls is 2? You may use approximations assuming 1000 is large.