Sleeping Beauty

by Stubborn Mule on 26 August 2014 · 0 comments

Sleeping BeautyFor the last couple of weeks, I have fallen asleep thinking about Sleeping Beauty. Not the heroine of the Charles Perrault fairy tale, or her Disney descendant, but the subject of a thought experiment first described in print by philosopher Adam Elga as follows:

Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

Elga, A. “Self‐locating belief and the Sleeping Beauty problem”, Analysis 60, 143–147 (2000)

It has become traditional to add that Sleeping Beauty is initially put to sleep on Sunday and is either woken up on Monday (Heads) or Monday and Tuesday (Tails). Then on Wednesday she is woken for the final time and the experiment is over. She knows in advance exactly what is going to take place, believes the experimenters and trusts that the coin is fair.

Much like the Monty Hall problem, Sleeping Beauty has stirred enormous controversy. There are two primary schools of thought on this problem. The thirders and the halfers. Both sides have a broad range of arguments, but put simply they are as follows.

Halfers argue that the answer is 1/2. On Sunday Sleeping Beauty believed that the chance of Heads was 1/2, she has learned nothing new when waking and so the chances are still 1/2.

Thirders argue that the answer is 1/3. If the experiment is repeated over and over again, approximately 1/3 of the time she will wake up after Heads and 2/3 of the time she will wake up after tails.

I first came across this problem myself on the blog of my former supervisor Bob Walters, who describes the thirder position as an “egregious error”. But as Bob notes, there are many in the thirder camp, including Adam Elga himself, physicist Sean Carroll and statistician Jeffrey Rosenthal.

As for my own view, I will leave you in suspense for now, mainly because I’m still thinking it through. Although superficially similar, I believe that it is a far more subtle problem than the Monty Hall problem and poses challenges to what it means to move the pure mathematical theory of probability to a real world setting. Philosophers distinguish between the mathematical concept of “probability” and real world “credence”, a Bayesian style application of probability to real world beliefs. I used to think that this was a bit fanciful on the part of philosophers. Now I am not sure sure: applying probability is harder than it looks.

Let me know what you think!

Image Credit: Serena-Kenobi

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John Carmody returns to the Mule in his promised second guest post and takes a close look at Australia Post’s profitability with some (ahem) back-of-the-envelope calculations.

There are many forms of communication which underpin the function and productivity of a modern society like Australia. Despite the Cassandra-commentary from Mr Ahmed Fahour (the well-paid CEO of Australia Post), regular mail delivery certainly remains one of them.

In making his tendentious, but opaque, points, he has not been entirely frank with the community. He has, for instance, claimed that 99% of our mail is electronic. That assertion is meaningless because so much e-mail is advertising, brief inter- or intra-office memos and notices, or quick substitutes for telephone calls. When these are removed from the calculation, the importance of “hard mail” becomes more obvious

The data which the Herald has published (for instance, “Please Mr Postman: snail mail doomed to disappear“, 14 June) also show how shallow or formulaic Mr Fahour’s thinking seems to be. In 2012-13 Australia Post made an after-tax profit of $312 million and if there had been no losses on the handling of letters, that would have been $530 million. Do Australians really want a profit of that magnitude from such a vital national service?

But when one looks at that “letter-loss” a little more closely and at the figure of 3.6 billion letters delivered that year, it is clear that the loss per letter was 6.5 cents. In other words, if instead of recently increasing the cost of a standard letter to 70 cents, this had been to 75 cents, the losses would have been comprehensively dealt with.

Some comparisons might be informative. The British Royal Mail currently charges about $A1.10 for delivery of a standard (20g) letter for next-day delivery within the UK (its “aim”) and $A0.95 if you’re happy for delivery within 3 days. The Deutsche Post charges the equivalent of 86 Australian cents for delivery within Germany but about $A1.08 cents to adjacent France. Given that we currently pay only 70 cents for delivery across a far larger area, my suggested price of 75 cents seems reasonable and justified.

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The government’s medical fairyland

14 June 2014

For the first time in a while, John Carmody returns to the Stubborn Mule with the first of two guest posts. He argues that the government’s proposed medical “co-payments” do not add up. The government continues to flounder about many details of its budget and part of the reason is a lack of stated clarity […]

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Government spending

20 May 2014

Before, during and after this month’s budget, Treasurer Joe Hockey sounded dire warnings about Australia’s “budget emergency”. Amidst this fear-mongering, it was a pleasant relief to come across a dissenting view. In a recent interview on 2SER Dr Stephanie Kelton (Department of Economics at the University of Missouri in Kansas City) argued that the government budget is very […]

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Randomness revisited (mathsy)

21 April 2014

My recent randomness post hinged on people’s expectations of how long a run of heads or tails you can expect to see in a series of coin tosses. In the post, I suggested that people tend to underestimate the length of runs, but what does the fox maths say? The exploration of the numbers in this post draws on […]

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Do Daleks use toilet paper?

18 April 2014

I have been watching some (very) old Doctor Who episodes, including the first ever serial featuring the archetypal villains, the Daleks. In this story, the Daleks share a planet with their long-time enemies, the Thal. After a war culminating in the denotation of a neutron bomb, both races experience very different mutations. The Daleks have […]

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Randomness

6 April 2014

With three children, I have my own laboratory at home for performing psychological experiments. Before anyone calls social services, there is an ethical committee standing by (their mother). This evening, I tried out one of my favourites: testing the perception of randomness. Here is the setup: I gave the boys two pieces of paper and […]

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Chinese non-residents…in China

31 March 2014

Recently I travelled to China for the first time. My first glimpse of Beijing took in the Escher-like headquarters of Chinese TV station CCTV. It is an extraordinary building and to get a proper sense of it, you have to see it from a number of different angles. Driving across the city, impressed by the […]

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Bringing Harmony to the Global Warming Debate

25 February 2014

For some time now, our regular contributor James Glover been promising me a post with some statistical analysis of historical global temperatures. To many the science of climate change seems inaccessible and the “debate” about climate change can appear to come down to whether you believe a very large group of scientists or a much […]

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I’m with Felix

16 February 2014

FT blogger Felix Salmon and venture capitalist Ben Horowitz have very different views of the future of Bitcoin. Salmon is a skeptic, while Horowitz is a believer. A couple of weeks ago on Planet Money they agreed to test their differences with a wager. Rather than a simple bet on the value of Bitcoin, the bet centres […]

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