z2newton
z2newton
Joined: Mar 22, 2014
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June 25th, 2015 at 10:38:19 PM permalink
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.
RS
RS
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June 25th, 2015 at 10:53:47 PM permalink
What are we trying to solve?
FleaStiff
FleaStiff
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June 26th, 2015 at 1:25:18 AM permalink
whether she should have a greater than even guess as to the proper answer to give.
Canyonero
Canyonero
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June 26th, 2015 at 7:08:38 AM permalink
She should always guess "tails". She will be right on Mondays and Tuesdays and will be wrong only on Mondays. 66% correct.
z2newton
z2newton
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June 26th, 2015 at 9:13:22 AM permalink
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.
RS
RS
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June 26th, 2015 at 11:37:05 AM permalink
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...
MangoJ
MangoJ
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June 27th, 2015 at 12:57:49 AM permalink
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.
SDSDNSR
SDSDNSR
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July 9th, 2015 at 8:03:33 PM permalink
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.

Whenever she wakes up, she flips a fair coin twice and requests the bet to be canceled unless the coin comes up heads twice. That is, she requests that the bet be canceled with probability 3/4. The expected value to her of the bet is $4,000, which you can easily prove is the highest possible value.
Zcore13
Zcore13
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July 9th, 2015 at 10:39:58 PM permalink
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
I am an employee of a Casino. Former Table Games Director,, current Pit Supervisor. All the personal opinions I post are my own and do not represent the opinions of the Casino or Tribe that I work for.
MathExtremist
MathExtremist
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July 9th, 2015 at 11:26:17 PM permalink
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%)
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563

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