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

Abstract

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.