A Puzzle About Information, Probability and Surprise

Abstract

Suppose 92 coins, flipped in succession, all come up heads. If we were previously confident that the process was fair, we would be surprised at this outcome. What, if anything, explains our surprise? And is it warranted? In what follows I do two things. First, I propose and defend an explanation of our surprise: we are surprised at the 92-head sequence, and various other sequences, because they are patterned. Second, Martin Smith (2017) has argued that, on an initial assumption that the coin-flipping process is fair, an observation of 92 heads does not warrant surprise. Against Smith, I argue that even if we knew beforehand that the coin-flipping process is fair, our knowledge is defeated by an observation of 92 heads. Under reasonable constraints on the prior probability that our initial assumption of fairness was wrong, an observation of 92 heads (or various other patterned outcomes) makes us practically certain that the process is unfair. As such, an observation of 92 heads does warrant surprise.