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### Transcript of Argentina Buenos Aires - UCEMA 2016. 5. 20.¢ UCEMA: Av. C£³rdoba 374, C1054AAP...

UNIVERSIDAD DEL CEMA

Buenos Aires

Argentina

Serie

DOCUMENTOS DE TRABAJO

Área: Probabilidades y Filosofía

Sleeping Beauty on Monty Hall

Michel Janssen y Sergio Pernice

Mayo 2016

Nro. 587

www.cema.edu.ar/publicaciones/doc_trabajo.html

UCEMA: Av. Córdoba 374, C1054AAP Buenos Aires, Argentina

ISSN 1668-4575 (impreso), ISSN 1668-4583 (en línea)

Editor: Jorge M. Streb; asistente editorial: Valeria Dowding

Sleeping Beauty on Monty Hall

Michel Janssen∗ and Sergio Pernice†

May 19, 2016

Abstract

We present a game show that we claim can serve as a proxy for the notorious Sleeping Beauty Problem. This problem has divided commentators into two camps, ‘halfers’ and ‘thirders’. In our game show, the potential awakenings of Sleeping Beauty, during which she will be asked about the outcome of the coin toss that determined earlier how many times she is awakened and asked, are replaced by potential contestants, deciding whether to choose heads or tails in a bet they will get to place if chosen as contestants on the outcome of the coin toss that determined earlier how many of them are chosen as contestants. This game show bears out the basic intuition of the thirders. Our goal in this paper, however, is not to settle the dispute between halfers and thirders but to draw attention to our game-show proxy itself, which realizes a version of the Sleeping Beauty Problem without the ambiguities plaguing the original. In this spirit, we design similar game-show proxies for variations on the Sleeping Beauty Problem with stochastic experiments other than a coin toss. We do the same for a variation in which Sleeping Beauty must decide upon being awakened whether or not to switch doors in the famous Monty Hall Problem and have the number of awakenings during which she gets to make that decision depend on the door she picked before she was put to sleep.

1 The Three Stooges on Monty Hall

Consider the following puzzle. 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 col- lectively pick one of three doors, D1, D2 or D3. Behind one are two checks for a

∗Program in the History of Science, Technology, and Medicine, University of Minnesota; email: janss011@umn.edu †Universidad del Cema, Buenos Aires; email: sp@cema.edu.ar

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thousand dollars each, behind the other two is a goat. The Three Stooges pick D2. Before the show goes to a commercial break, Monty tells the Three Stooges that either one or two of them will be called back after the break. If they were wrong, only one of them will return (but he doesn’t tell them which one!); if they were right, the other two will. During the break, the Three Stooges are made to take a nap backstage. When the show resumes, they are sound asleep.

If the checks are not behind D2, Monty wakes up Curly and brings him back to the stage, making sure he has no idea whether he is the first or the second one to be woken up and brought back. As usual, 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. Curly is given ample time to make up his mind. If the door he ends up choosing has the two checks behind it, he gets one of them. If not, he goes home empty-handed.

If the checks are behind D2, Monty goes through this same routine twice, with Moe and Larry (not necessarily in that order), the only other difference being that he now has a choice whether to open D1 or D3. Monty once again makes sure that neither of them finds out whether they were woken up and brought back first or second (at least not until Monty opens the door with the checks behind it, one of which may be gone at that point).

What is a Stooge to do in this predicament? Should he switch? Should he stay with the door they originally picked? Does it make any difference whether he stays or switches?

2 Game-show proxies for (variations on) the Sleep-

ing Beauty Problem

The 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.1 The solution of the Monty Hall Problem is no longer controversial. A contestant in a normal episode of “Let’s make a deal” (without the funny business of putting contestants to sleep and waking them up again) should always take Monty Hall up on his offer to switch. Since Monty Hall never opens the door with the prize behind it and thus 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 (weak) assumption is routinely granted.

1There is a vast literature on both. Good places to start are the wikipedia entries for these two problems.

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The intuition that the opening of a door and the offer to switch are just for dramatic effect and do not affect the contestant’s chances of winning is simply wrong. The opening of one of the doors provides the contestant with important information. A simple and (judging by its ubiquity on the web) effective way to make the point is this. Initially, the contestant only has a 1/3 chance of picking the right door and a 2/3 chance of picking the wrong one. Suppose he (or she) switches. If he was right the first time, he will now be wrong. If he was wrong the first time, he will now be right. So he now has a 2/3 chance of being right and only a 1/3 chance of being wrong. In a slogan, he flips the odds by switching. A device often used to shore up one’s intuitions in this case is to increase the number of doors. If there are n doors (n ≥ 3), the contestant initially has a 1/n chance of picking the right one and a (n − 1)/n chance of picking the wrong one. After the contestant has made her (or his) initial choice, Monty Hall opens all but one of the remaining doors and offers her to switch. Once again, the contestant flips the initial odds by switching. By switching, she effectively guesses that the prize is not behind the door she initially picked but behind any one of the other n− 1 doors, all but one of which Monty Hall has meanwhile opened for her.

Unlike the Monty Hall Problem, the Sleeping Beauty Problem remains contro- versial. The problem is essentially the following. Sleeping Beauty is told that a fair coin will be tossed after she’s been put to sleep and that, when she is woken up, she will be asked what her degree of belief is that the coin came up heads. It depends on the outcome of that coin toss, however, how many times she is asked. If the coin comes up heads, she will only be woken up and asked once. If the coin comes up tails, she will be woken up and asked twice. The first time she’s woken up after the coin comes up tails she is given some amnesia drug so that she won’t remember the second time that she’s been woken up before and asked the same question. Every precaution is taken to make sure that Sleeping Beauty, when she wakes up, cannot tell whether her current awakening is the one after the coin came up heads or one of the two after the coin came up tails. What should Sleeping Beauty’s degree of belief be upon being awakened that the coin came up heads?

One knee-jerk response is that, no matter how often Sleeping Beauty is put to sleep, woken up and drugged, the probability that a fair coin comes up heads is and remains 1/2. Therefore, her answer every time she is asked should be 1/2. Another knee-jerk response is that, if the coin comes up tails, Sleeping Beauty is twice as likely to be asked than if the coin comes up heads. Therefore, her answer every time she is asked should be 1/3. Those who think the answer is 1/2 are known as halfers. Those who think the answer is 1/3 are known as thirders. The debate between halfers and thirders has long moved beyond this clash of intuitions upon first encountering

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the problem. In fact, it persists only because it is ambiguous exactly what Sleeping Beauty is being asked.

To get a better handle on the Sleeping Beauty Problem and inspired by the Monty Hall Problem, we design a game-show proxy for it, in which potential contestants take over the role of potential awakenings. Our analysis will show that both halfers and thirders are right, depending on how one interprets the question Sleeping Beauty is asked. That said, our game-show proxy will be much more congenial to thirders than to halfers as it implements the interpretation of the question they consider to be the interesting one. Should the information that Sleeping Beauty is given ahead of time about what will happen depending on the outcome of one toss of a fair coin change her degree of belief upon being awakened that this particular coin toss resulted in heads? Upon a little reflection, both halfers and thirders will agree that if that is the question, Sleeping Beauty would be just as mistaken to ignore this information as contestants on “Let’s make a deal” would be to ignore the information Monty Hall is giving them by opening one of the three doors.

Just as Sleeping Beauty knows how a coin toss will decide how many of her potential awakenings will become actual awakenings, potential contestants in our game-show proxy know how a coin toss will decide how many of them become actual contestants. Both Sleeping Beauty and our potential contestants are asked to assess (one way or another) the probability that this coin toss resulted in heads when woken up or chosen as a contestant, respectively. Using Bayes’ rule, they shou