A bit like the Boy-Girl problem...but much more slippery*
12 February 2017
A week ago I'd never heard of the Sleeping Beauty puzzle. Since then two people have asked me if I know it, so I can only guess that it is currently doing the rounds.
The puzzle - which is over 30 years old - is very contrived, but I'll share it anyway because it has apparently divided probability theorists into two camps, and I don't think it will be long before somebody else asks me for my opinion on it.
(Note: I have added Prince Charming to the original version, to give the story a more tangible outcome.)
THE SLEEPING BEAUTY PROBLEM
It is Sunday evening, and The Luck Fairy has told Sleeping Beauty that Prince Charming wants to come on Tuesday lunchtime and take her to his castle. Whether he turns up or not will depend on the flip of a coin.
The Luck Fairy explains to Sleeping Beauty: "Tonight I will give you a sleeping potion. Then I will flip a coin. If the coin comes up Heads, you will sleep through until Tuesday morning when I will wake you. Prince Charming will arrive for you later.
If the coin comes up Tails,I will send a message to Prince Charming telling him not to come. I will wake you up on Monday morning. Then I will give you another sleeping potion which wipes any memory of the fact that you woke up on Monday. On Tuesday morning I will wake you up again."
Sleeping Beauty falls asleep. She is in a sealed room with no way of knowing whether it is Monday morning or Tuesday morning. She is woken by the Luck Fairy who asks her: "What do you think is the chance that Prince Charming will be coming to take you to the castle?"
Her first thought is that the chance is clearly 50:50. The coin either came up Heads or Tails, so the odds on Prince Charming coming are evens. And that's what I thought the answer was too, when I first heard this puzzle. The odds for Prince Charming were 1/2 at the start of the exercise, and when Sleeping Beauty wakes she has no new information so the odds can't have changed.
However there is an argument for a different answer.
When Sleeping Beauty wakes, it is one of three situations: it could be Monday morning (Tails), Tuesday morning (Heads) or Tuesday morning (Tails). And these three instances are equally likely. How come? If we were to repeat this experiment 30 times, Sleeping Beauty would on average be woken 15 times on a Monday because the coin came up Tails, 15 times on a Tuesday when it's Tails and 15 times on a Tuesday when it's Heads. When she wakes up she has no idea which of these 45 instances it is, and only in 15 of the 45 instances did the coin come up Heads. Therefore she can reason that the chance that Prince Charming will come is only one in three.
One way to resolve questions like this is by working out what would be a 'fair' bet for Sleeping Beauty to make on Prince Charming coming. Let's suppose she believes that the odds that Prince Charming will be coming for her on Tuesday are indeed 1/2. And let's imagine his arrival is worth £6million to her, and she wants to bet on him coming. With odds of 1/2, she can stake £3million on him coming and on average she won't lose out (half the time she'll lose £3m, half the time she'll end up with £3m profit).
Suppose Sleeping Beauty always bets when given a chance. Because she believes that the odds are 1/2, she will bet £3m. If the coin comes up Heads, she bets £3m when she wakes on Tuesday. If the coin comes up Tails, she bets £3m when she wakes on Monday, and (having of course forgotten everything from the day before) she then bets another £3m when she wakes on Tuesday. So if the coin ends up Heads she wins a net £3m (£6m for Prince Charming minus her £3m bet). But if it comes up Tails, she bets £3m twice, and loses £6m. In other words, if she thinks the odds are 1/2 she ends up losing money on average, which means it isn't a fair bet.
Only by betting £2m each time can she expect to end up evens. And a £2m bet for a potential £6m payout means that she should regard the probability as being 1/3.
* * *
I said at the start that probability theorists are divided into two camps. They are known as the 'halfers' and the 'thirders'. Actually there's another group - those who reject this as a puzzle that can be analysed by probability theory. And there's a fourth group too, the ones who say: "Get a life."
* Thanks to @realityminus3 for pointing out that Cinderella is more slippery than Sleeping Beauty