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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.

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Given the difficulties of defining “machine” and “think”, Turing proposed to replace the question “Can machines think?” with a proxy: how well can an agent engage in sustained conversation with a human? Though Turing neither described himself as a philosopher nor published much on philosophical matters, his Imitation Game has stood the test of time. Most understood at that time that success would not come easy, but few would have guessed just how difficult engaging in ordinary conversation would turn out to be. Despite the proliferation of language processing tools, we have seen little progress towards doing well at the Imitation Game. Had Turing instead suggested ability at games or even translation as a proxy for intelligence, his paper might have been forgotten. We argue that these and related problems are amenable to mechanical, though sophisticated, formal techniques. Turing appears to have taken care to select sustained, productive conversation and that alone as his proxy. Even simple conversation challenges a machine to engage in the rich practice of human discourse in all its generality and variety.