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CTK Exchange
william
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Apr-14-03, 12:37 PM (EST) |
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"sleeping beauty"
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Hi,
i'd like to discuss Sleeping Beauty (SB) puzzle:
you flip a coin;
heads you wake SB on monday and make her a question.
tails you wake SB on monday and make her the same question, THEN you make her sleep again and forget everything, THEN you wake her on tuesday and make her the same question.
The question you make is: what is the probability the coin turned heads ?
OBS: she never knows which day she is and if is tuesday she doesn't remember she woke on monday.
I,ve seen two answers: 1/2 and 1/3 and both have good arguments.
any ideas ?thanks |
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Whymme
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Apr-14-03, 06:05 PM (EST) |
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1. "RE: sleeping beauty"
In response to message #0
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I guess that there is some information lacking from this problem. As it is stated now, it doesn't matter what you do with the sleeping beauty; if the coin is honest there is a 50% chance.
Also note how the question changes from present tense ("you flip a coin" to past tense ("(what is the probability that) the coin turned heads?". That doesn't make understanding the problem any easier.A similar question would be: "Harry flips a coin; if it's heads he orders pizza and if it's tails he orders chinese. What is the probability that the coin turned heads?" Whymme |
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william
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Apr-15-03, 08:38 AM (EST) |
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2. "RE: sleeping beauty"
In response to message #1
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I'm sorry if the puzzle is not clear. It may be because english is not my native language.
Anyway, the coin is fair, so the chances are 50%, no doubt.
What makes it different from your Harry example is that if the coin turns tails you wake SB TWICE, but she doesn't know it. So, when she wakes she doesn't know what day she is.
If the coin turned heads(when she wakes the coin turned in the past)it must be monday when she wakes, but if it turned tails it may be monday or tuesday.
I hope it is clearer nowWilliam
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Whymme
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Apr-18-03, 06:04 PM (EST) |
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5. "RE: sleeping beauty"
In response to message #2
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Well, English isn't my native language either; we therefore should be able to understand each other completely  Thanks to Steve's exposition below I now understand the problem a bit better. It looks like "1/2" and "1/3" are answers to different questions - or at least the same question to different actors in the set-up. My answer of 50% was given as the bystander who sees the coin being tossed and doesn't care what happens as a result of that toss - see my answer of Harry's menu choices. Steve's answer was given from the POV of the Sleeping Beauty. And then you indeed get a different answer. This is probably the point of view that you meant, but I missed that (it would have been easier to understand if the "you" in the problem had been the Sleeping Beauty, not the person who wakes her). Anyway, Steve is correct. There are three possibilities, all three equally likely:
- It is Monday and the coin came up heads
- It is Monday and the coin came up tails
- It is Tuesday and the coin came up tails There are certainly parallels to be drawn with the Monty Hall problem, so I can imagine that this question will generate a long discussion. Let me try to explain the matter in my own words: You are the SB. You're awoken and asked the question. So you start thinking:
- It's either Monday or Tuesday.
- The chance of this being Tuesday is only half as large as this being Monday; it can only be Tuesday if the coin landed 'tails'.
- So the chance that this is Tuesday is 1/3, the chance that this is Monday is 2/3.
- However, *if* this is Tuesday, the chance that the coin landed 'tails' is 100%.
- OTOH, if this is Monday, there is only a 50% chance that the coin landed tails, and a 50% chance that the coin landed heads.
Therefore, the chance of the coin having landed tails is (1/3)*100% + (2/3)*50% --> for a total of 2/3. Whymme |
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SteveSchaefer
Member since Apr-2-02
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Apr-15-03, 08:38 AM (EST) |
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3. "RE: sleeping beauty"
In response to message #0
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You are asking Sleeping Beauty the question: "What is the probability that the coin flip was heads?" If the coin is heads, then she is asked the question once (on Monday) If the coin is tails, then she is asked the question twice (on Monday and again on Tuesday). The easiest way to do this type of problem (conditional probability) is to add up all of the possibilities, as long as they are all equal probability. There are three possible questions, and each question is asked with the same probability (0.50). The average number of questions asked is 1.5. Only one of the three questions occurs after a heads flip, so the answer is 1/3. A more careful way to solve this is to break out the conditional probabilities. The question that you are trying to ask is "Given that I am asking a question today, what is the probability that the coin flip was heads?" There are four possibilities to consider. Heads/Monday, Heads/Tuesday, Tails/Monday, Tails/Tuesday. Each of these has a probability of 1/4, but a question is not asked for the Heads/Tuesday case. P(Heads) = P(Heads & Question) + P(Heads & no Question)
= P(Question)*P(Heads|Question) + P(no Question)*P(Heads|no Question) where P(Heads|Question) is read "probability of Heads given Question." This is the quantity that we want. P(Heads) is 1/2; P(Question) is 3/4, P(no Question) is 1/4. P(Heads|no Question) is 1, because the only time that Sleeping Beauty isn't asked a question is if the coin flip was heads. Alternatively, you could recognize that P(Heads & no Question) is 1/4. 1/2 = (3/4)*P(Heads|Question) + 1/4
1/4 = 3/4*P(Heads|Question)
P(Heads|Question) = (1/4)/(3/4) = 1/3 |
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william
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Apr-16-03, 02:01 PM (EST) |
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4. "RE: sleeping beauty"
In response to message #3
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Though i'm not expert in statistics I tend to agree with 1/3 answer, but there are some good points for 1/2 also: >There are three possible questions, and each question is asked with the same probability That seems to be a tricky point for people who favors 1/2 answer: the treee events are not of equal probability because "wake tuesday" onle occurs if "wake monday" ocurrs, so it depends on the first event. The argument goes to the point where in fact you have only two possibilities. In fact i just wanted to raise the issue and learn from different points of view. I can't truly defend or atack any answer, for i don't have the tools for that william |
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Graham C
Member since Feb-5-03
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Apr-26-03, 12:59 PM (EST) |
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6. "RE: sleeping beauty"
In response to message #0
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There are only two possible outcomes: a) It was heads and you asked the question once on Monday
b) It was tails and you asked the question twice, on Monday and Tuesday Each has probability 0.5. How Sleeping Beauty might answer the question you can't tell, because you don't know what she knows about probability. Probably she'd say 'What?'. |
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