One game show, two boys, two aces, three prisoners - what’s an AI to do?

We review a quartet of widely discussed probability puzzles – Monty Hall, the three prisoners, the two boys, and the two aces. Pearl explains why the Monty Hall problem is counterintuitive using a causal diagram. Glenn Shafer uses the puzzle of the two aces to justify reintroducing to probability theory protocols that specify how the information we condition on is obtained. Pearl, in one treatment of the three prisoners, adds to his representation random variables that distinguish actual events and observations. The puzzle of the two boys took a perplexing twist in 2010. We show the puzzles have similar features, and each can be made to give different answers to simple queries corresponding to different presentations of the word problem. We offer a unified treatment that explains this phenomenon in strictly technical terms, as opposed to cognitive or epistemic.

Sleeping Beauty on Monty Hall

Inspired by the Monty Hall Problem and a popular simple solution to it, we present a number of game-show puzzles that are analogous to the notorious Sleeping Beauty Problem (and variations on it), but much easier to solve. We replace the awakenings of Sleeping Beauty by contestants on a game show, like Monty Hall’s, and increase the number of awakenings/contestants in the same way that the number of doors in the Monty Hall Problem is increased to make it easier to see what the solution to the problem is. We show that these game-show proxies for the Sleeping Beauty Problem and variations on it can be solved through simple applications of Bayes’s theorem. This means that we will phrase our analysis in terms of credences or degrees of belief. We will also rephrase our analysis, however, in terms of relative frequencies. Overall, our paper is intended to showcase, in a simple yet non-trivial example, the efficacy of a tried-and-true strategy for addressing problems in philosophy of science, i.e., develop a simple model for the problem and vary its parameters. Given that the Sleeping Beauty Problem, much more so than the Monty Hall Problem, challenges the intuitions about probabilities of many when they first encounter it, the application of this strategy to this conundrum, we believe, is pedagogically useful.