Sleeping Beauty Problem

I was reading a blog where it talked about this probability puzzle. I'd never heard of the Sleeping Beauty problem before so I thought I'd post it here to see what the more math inclined have to say about it. From what I can gather there is not a consensus on the answer.

[http://en.wikipedia.org/wiki/Sleeping_beauty_problem]


Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed, she is asked, "What is your belief now for the proposition that the coin landed heads?"

I guess they knock her out before she can ask what day it is.

[edit]
Since this is a gambling forum here is an interesting twist I found. The experiment is the same except Sleeping Beauty is given the chance to make a bet before the experiment begins. She is given 3 to 2 odds that the coin will come up heads. Instead of the original question, when she is awakened she is asked if she wants to cancel her bet.

To make it interesting lets say she bets $1000 on heads.

What are we trying to solve?

whether she should have a greater than even guess as to the proper answer to give.

She should always guess "tails". She will be right on Mondays and Tuesdays and will be wrong only on Mondays. 66% correct.

Quote: Canyonero

She should always guess "tails". She will be right on Mondays and Tuesdays and will be wrong only on Mondays. 66% correct.

Heads is 1/3? So it's like the monty hall problem.

Not like monty hall. Well maybe?

But if it's tails, then you'll get interviewed more often than if it's heads.


I can't wait until Alan and his dice crew start saying it's 50/50 because there are 2 sides of a coin...

Quote: z2newton

I guess they knock her out before she can ask what day it is.

What is your question anyway ?

If she gets the interview and doesn't know anything about previous interviews or the day, 2/3 will be tails and 1/3 will be heads.
You can solve all those questions with Bayesian inference.

Let's take a specific example. On Sunday Sleeping beauty is offered to bet on a coin flip. If the result is heads, she wins $64,000. If it is tails, she loses $128,000. She will not learn the result, however, until after she has gone through the sleeping and waking process. If the result is heads, she will be put to sleep and awakened on Monday and asked if she would like to cancel the bet, then put back to sleep until Wednesday. If the result is tails, she will be put to sleep and awakened on Monday and asked if she would like to cancel her bet, given the amnesia inducing drug, put back to sleep, and awakened on Tuesday with the same question about canceling the bet. Then she will be awakened on Wednesday and the bet resolved.

If the result of the coin toss is tails, she is queried twice, the bet being canceled if she so requests on either or both occasions and remaining active if she does not cancel it on either occasion. And of course if the result is heads and she requests cancellation on Monday, it is also canceled.

What strategy can she use to create the largest possible positive expected value of the bet, and what will be that expected value? Because of the amnesia, she must use the same strategy every time she wakes up.

It does not say who is flipping the coin.

Is is possible Sleeping Beauty previously spent 7 years practicing coin flipping influence and maybe even attended a class and can now know the outcome based on how she holds the coin prior to flipping? Or did she get to choose the person flipping the coin as a show of fairness, in which case she would obviously choose the famous Coin Flipping Captain.


ZCore13

Quote: Zcore13

It does not say who is flipping the coin.

Is is possible Sleeping Beauty previously spent 7 years practicing coin flipping influence and maybe even attended a class and can now know the outcome based on how she holds the coin prior to flipping?

http://statweb.stanford.edu/~susan/papers/headswithJ.pdf
(Summary: when flipping a coin heads-up, the chance of heads is about 50.8%)