Abstract
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.
Highlights
Monty Hall and Sleeping BeautyThe puzzle in Section 1 combines elements of two well-known puzzles challenging our intuitions about probability: the Monty Hall Problem and the Sleeping Beauty Problem.2 The solution of theMonty Hall Problem is no longer controversial
Because Monty Hall never opens the door with the prize behind it and, has to know which door that is, we need to assume that he offers contestants to switch regardless of which door they initially picked, but that assumption is routinely granted
Sleeping Beauty and our potential contestants are both asked to assess the probability that this coin toss resulted in heads when woken up or selected as a contestant, respectively
Summary
In a special edition of his famous game show, “Let’s Make a Deal”, Monty Hall calls the Three Stooges to the stage and has them collectively pick one of three doors, D1 , D2 or D3. Before the show goes to a commercial break, Monty tells his special guests that either one or two of them will be called back after the break. Monty opens either D1 or D3 (whichever one has a goat behind it) and offers Curly to switch from D2 to the other door that remains unopened. If the door he chooses has the two checks behind it, he gets one of them. What is a Stooge to do in this predicament? Should he switch? Should he stay with the door they initially picked? Should he be indifferent between staying and switching?1
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