The Sleeping Beauty Controversy

I am reading “The Best Writing on Mathematics 2018” and amongst the various fascinating articles is one on “The Sleeping Beauty Controversy” by Peter Winkler.

the problem

(Here’s a video link – which examines the problem but isn’t related directly to Peter Winkler’s piece)

To steal Professor Winkler’s own description from the article:

Sleeping Beauty agrees to the following experiment. On Sunday she is put to sleep, and a fair coin is flipped. If it comes up heads, she is awakened on Monday morning; if tails, she is awakened on Monday morning and again on Tuesday morning. In all cases, she is not told the day of the week, is put back to sleep shortly after, and will have no memory of any Monday or Tuesday awakenings.

When Sleeping Beauty is awakened on Monday or Tuesday, what – to her – is the probability that the coin came up heads?

The bulk of mathematicians are “thirders” – they think the correct answer is 1/3. But a sizable minority are “halfers” – believing the correct answer is 1/2. Still others think the question is undecidable.

For what it’s worth I am with the thirders – Sleeping Beauty would understand that if, for instance, the coin was tossed 100 times she’d be woken around 150 times and so the chances she’s being woken as a result of a head are one-in-three.

The problem restated?

But what really struck me in the article was what Professor Winkler says is a rephrasing (though admits some might disagree) of Sleeping Beauty – because this leads me to a “halfer” position:

Alice, Bob and Charlie have each taken a new sleeping pill. In hospital experiments half the subjects slept through the night… but the other half woke up once in the middle of the night then returned to sleep and woke up in the morning with no memory of the night awakening.

Alice wakes up in the middle of the night… her credence that the pill has worked drops to zero… Bob wakes up in the morning, his credence that the pill worked remains at 1/2…

Charlie, who has blackout shades in his bedroom, wakes up not knowing whether it is morning. According to the thirders, his credence in the efficacy of the pill is 1/3 until he raises the shades…

But how can this be? Charlie has no additional information beyond that he has woken up (as he would do in any case) and so surely his prior information – that the pill is 1/2 effective remains unchanged?

(Here’s another link to a different article on this question.)

A bit of fun: Shawn Urban’s math puzzle

Before you read the rest, maybe you should have a look at Shawn’s blog… as what’s below may spoil the puzzle.

He writes:

Every math week, I plan to provide a math challenge which could take you seconds to hours to solve, assuming you don’t cheat by using technology to solve it for you.

Today, I offer a time filler I occasionally use while substituting a junior or senior high math class that has “completed” all of its work, usually before I even enter class. I learned this problem from Blaine Dowler of Sylvan Learning. Usually I reward the first person who solves it, and usually it is the “weakest” student who does it.

The challenge is: If the product of two numbers is two and their sum is five, what is the simplest sum of their inverses?

I think I understand why the “weakest” student answers first, because the naive answer is the correct one:

Let our numbers be $m$ and $n$. So:

$mn = 2$ and $n + m = 5$

We are looking to solve $\frac{1}{m} + \frac{1}{n}$

$\frac{1}{m} + \frac{1}{n} = \frac{n + m}{nm} = \frac{5}{2}$

Have to say this did not occur to me immediately at all, though I could hear a faint voice telling me to try it for some time before I actually did.