Sleeping Beauty Paradox

[KipBond]

[ QUOTE ]
She knows that if she takes two Week long trips she will get on average 3 awakenings, one due to heads and two due to tails. But should she think the three of them are equally likely to be the one she is experiencing right now? No. <font color="red">She is more likely to be exeriencing the one due to heads! She knows there is a 50% chance she is experiencing That one.</font> There is only a 25% chance she is experiencing the one due to tails on monday. And only a 25% chance she is experiencing the one due to tails on tuesday. Just because there are three types of awakenings doesn't mean they are equally likely to be the one she is experiencing right now.

[/ QUOTE ]

This can't be right. So, if we tell her it is Monday, then there is a 2/3 chance the coin was heads? I don't see how. It has to be 1/2:

P(H|monday) = 50%
P(H|tuesday) = 0%
P(T|monday) = 50%
P(T|tuesday) = 100%

P(monday|H) = 100%
P(tuesday|H) = 0%
P(monday|T) = 50%
P(tuesday|T) = 50%

If any of those are not right, I'd like do know why. It seems intuitive to me.

Now:

P(monday|waking) = 2/3
P(tuesday|waking) = 1/3

I take it this is where the disagreement/misunderstanding is. Because if you agree with those, then P(H|waking) &amp; P(T|waking) are pretty simple to compute.

[Deorum]

I did not read all of the posts, but I do not see a paradox.

To the question of, "What is the probability that you were awakened by a tails-flip?" the answer is 2/3. This is the equivalent of asking, "What day is it?" If she answers Monday, she will be correct 2/3 of the time, if she answers Tuesday, she will be correct 1/3 of the time. To illustrate this, imagine that we run the experiment 100 times (and for the sake of simplicity, let's fix the flip number at 50 heads and 50 tails). 50 times she will wake on Monday because of heads. 50 times she will wake on Monday because of tails. 50 times she will wake on Tuesday because of tails. If she answers Monday, she will get 100 correct and 50 wrong. If she answers Tuesday, she will get 50 correct and 100 wrong.

But this is not the same as saying that there was a 2/3 chance that the coin came up tails. The number of times she awakens due to a flip has nothing to do with whether or not it came up heads or tails. Rather, it has to do with the precedent set at the beginning of the experiment that she will wake twice when the coin was flipped tails and once when the coin was flipped heads.

She knows that she will wake twice as often on a tails flip than she will on a heads flip. She knows that her awakening is 1/3 likely due to a heads flip, and 2/3 likely due to a tails flip. But the awakenings that occur on Monday from a tails flip and the awakenings that occur on Tuesday from a tails flip are not derived from independent flips. Each Monday-Tuesday pair is due to one flip. In other words, 1 heads-Monday = 1 tails-Monday + 1 tails-Tuesday.

She knows all of this from the beginning. Since she knows that each tails flip results in her waking twice as often as a heads flip, she knows that there were only half as many "tails flips" as "tails awakenings". She also knows that the amount of "heads flips" and "heads awakenings" are equivelant. Therefore, the amount of "tails flips" and "heads flips" are equivalent, because:

.5(2/3) = 1/3

Since the amount of "heads flips" and "tails flips" are equivalent, she will state that the probability of the coin being flipped heads or tails is 50% for each.

Q.E.D.


Quick Proof:

She knows:

1 tails flip = two awakenings

1 heads flip = one awakening

Through substitution, 1 tail flip = 2 heads flips

She will have 2/3 awakenings due to tails, and 1/3 due to heads (on average - remember, she knows that).

Since each "tails awakening" is only worth half of a "heads awakening" in terms of flips, and there are twice as many "tails awakenings" than "heads awakenings" she will come to the conclusion that the coin comes up 50% heads and 50% tails.

[PairTheBoard]

First, let me simply restate my full argument, then respond to your points. Beauty has an information based Credence-probability function P before she enters the experiment. When she finds herself Awakened she considers if she should adopt a new information based Credence-probability function Q. She asks herself, "Do I have any new information?". She decides No. The information she had for forming P included the fact that she would now be in this situation. So she concludes that her Q should remain P. She let's Q=P and starts to think what that means.

She decides the probability the coin landed heads is still 50%. Just because she is now experiencing one of the three kinds of awakenings she knew beforehand she would be experiencing doesn't add any new information for how the coin might have landed. So Q(heads)=50%. That means she must conclude there is a 50% chance that this awakening is a monday heads awakening. Similiarly she reasons there is a 25% chance that this awakening is a monday tails awakening or tuesday tails awakening.

But she is now confused. That means she should bet $1 at even money that heads landed. But, she thinks, if she were to do that everytime she awakens in this experiment she will lose $1 per experiment. She wonders, should she change her Credence function Q so that it will give her the odds she needs to break even on these bets on heads?

Then she realizes something. When heads comes up she is only allowed to bet once. When tails comes up she is forced to bet twice. She thinks, that's not a legitimate bet. That's like offering me to bet even money on heads but telling me I must bet $1 when heads comes up and $2 when tails comes up. I don't change my estimate for the probability that heads will come up. I just don't take the bet. So Beauty stays with her Prior Credence function and just refuses the unfair bet on heads. She knows, that for this experiment there is a 50% chance the coin came up heads and if it did this must be monday.
=======================

Now to your points.

[ QUOTE ]

P(Coin is Heads &amp; I then wake her on Monday) = 50%
P(Coin is Tails &amp; I then wake her on Monday) = 50%
P(Coin is Tails &amp; I then wake her on Tuesday) = 50%

If you want to combine these events to make a total "event space" that totals 100%, then each "flip-waking" event has a 1/3 chance of happening.


[/ QUOTE ]

You have 3 Events there. The first is just the Event that the Coin lands heads. Both the second and third are the same event. They are the Event that the coin lands tails.

Beauty keeps those probabilities. They tell her that this experiment has a 50% chance of Heads. So if heads flipped this must be monday. If tails flipped she applies indifference to conclude it's equally likely to be monday or tuesday. Thus the Beauty's Event, "I am awake", has properly divided probabities to add to 100%.

[ QUOTE ]
After 1,000,000 coin flips:
~500,000 will be heads
~500,000 will be tails

and we will wake Beauty ~1,500,000 times:
~500,000 will be heads-wakings (coin was heads &amp; we woke her up on Monday)
~1,000,000 will be tails-wakings (coin was tails &amp; we woke her up on Monday &amp; Tuesday)

[/ QUOTE ]

Beauty is well aware of this fact. That's why she won't use her correct Credence of 50% that Heads landed this experiment to bet on heads at even money. She can see that the situation you describe above implies that the Outcome of the Bet will be Forcing Beauty to bet twice in losing situations and only allows her to bet once in winning situations.




You do not change your Probability for an outcome because there is a betting situation where the outcome of the bet dictates to you how much you will bet.


PairTheBoard

[KipBond]

[ QUOTE ]
"What is the probability that you were awakened by a tails-flip?" the answer is 2/3.
...
Since the amount of "heads flips" and "tails flips" are equivalent, she will state that the probability of the coin being flipped heads or tails is 50% for each.

[/ QUOTE ]

So, after being awakened, what is her answer to the question: "What is the probability that the coin was heads?"

The question is ambiguous. You give answers to both interpretations above.

[PairTheBoard]

Consider the following analogy. You have a computer with a good algorithm for processing information to produce a Credence-probability for Events defined by that information. You input into the computer the following information. You are going to flip a fair coin. The computer must output a Probability that the coin has landed heads. But you also tell it that if the coin lands Heads the computer will be outputing its computed probability for one day. If it lands Tails the computer will be outputing its computed probability for one year. But the computer will not know which of these conditions it is in because with Heads, its processor will be sped up by a factor of about 365. It will have no way of distingishing between the normal and sped up processing speed.

A signal will be given to the computer telling it to begin giving its output for the probability the coin landed heads. We will also ask it to give it's Credence-probability for What State it is In. The computer must output it's Belief probability for its being in a Sped up Processing State or a Normal Processing State. We tell it we will measure the number of real days it holds the correct Belief probability for the State it is in.

You also tell the computer that we will be taking its outputs and each real day we will bet 1 computer measuring unit on heads at the odds determined by its output for the probability that heads has landed. But we don't tell it how that should affect its output of the probability that heads has landed. We don't tell it to optimize anything with respect to those bets we are making. We just tell it to take all this information and compute the best Credence-probability that the coin landed heads, and its Belief probability for its State. Do this when we say Begin.



You have input that information. You then flip a coin, set the computer's processor speed accordingly and ask it to Begin giving outputs for the probability the coin landed heads. Let's suppose it outputs a 50% probability the coin landed on heads.

Now, are we going to criticize the computer because it believes there is a 50% chance the coin landed heads for only a day when the coin did land heads, and it believes the 50% for a year when the coin landed tails?

Are we going to criticize the computer because if we bet $1 on heads at odds determined by the computer's output of 50%, and we make 1 bet a day for every day the computer is giving that output we will lose money?

No. The computer is correct to give the output of 50%. The fact that it believes 50% longer for tails than for heads does not make it an incorrect belief.

PairTheBoard

[KipBond]

[ QUOTE ]
She decides the probability the coin landed heads is still 50%. Just because she is now experiencing one of the three kinds of awakenings she knew beforehand she would be experiencing doesn't add any new information for how the coin might have landed.

[/ QUOTE ]

I disagree. See my "new experiment" above where I only wake her when the coin is tails. Waking her DOES give her more information about the previous outcome of the coin she's being asked about.

[ QUOTE ]
the Outcome of the Bet will be Forcing Beauty to bet twice in losing situations and only allows her to bet once in winning situations.

You do not change your Probability for an outcome because there is a betting situation where the outcome of the bet dictates to you how much you will bet.

[/ QUOTE ]

That's about what I said in my 1st &amp; 2nd posts in this thread(*). I still agree. However --- I now think that because the question being asked Beauty is ambiguous, one of the ways she can answer the question allows her to use the new information she has when she is woken up.

(*) I then brought up why the question in this thread was not as ambiguous as the question posed in the website link you provided, and why "1/3" can be a valid answer to the more ambiguous wording of the question posed to Beauty.

[KipBond]

PTB: Please tell me where you disagree with the following:

P(H|monday) = 50%
- (Probability the coin that was flipped prior to Beauty's waking is Heads, given that today is Monday)
P(H|tuesday) = 0%
P(T|monday) = 50%
P(T|tuesday) = 100%

P(monday|H) = 100%
- (Probability that today is Monday, given that the coin we flipped prior to waking Beauty is Heads)
P(tuesday|H) = 0%
P(monday|T) = 50%
P(tuesday|T) = 50%

P(monday|waking) = 2/3
- (Probability that today is Monday, given that we are waking Beauty up in accordance with the outlined experiment parameters)
P(tuesday|waking) = 1/3
P(wednesday|waking) = 0

P(waking|monday) = 100%
- (Probability that we are waking Beauty up in accordance with the outlined experiment parameters, given that today is Monday)
P(waking|tuesday) = 50%
P(waking|wednesday) = 0%

[KipBond]

Sorry for posting so much... but I remembered that there was something in your first reply that I disagreed with:

[ QUOTE ]
So having awoken she knows there is a 75% chance it is Monday, and a 25% chance it is Tuesday.

[/ QUOTE ]

There are 3 flip+waking events, all equally likely:
1) Heads+Monday+Waking
2) Tails+Monday+Waking
3) Tails+Tuesday+Waking

If you ran this experiment 1,000,000 times there would be:
500,000 Heads+Monday+Wakings
500,000 Tails+Monday+Wakings
500,000 Tails+Tuesday+Wakings

So, when she wakes up, she knows that there is a 2/3 probability that it is Monday, and a 1/3 probability that it is Tuesday.

[ QUOTE ]
2/3 of the time that it's Monday will be because of Head. 1/3 of the time that it's Monday will be because of Tails. And all the time that it's Tuesday will be because of Tails.

[/ QUOTE ]

50% of the time that it's Monday will be because of Heads. 50% of the time that it's Monday will be because of Tails. 100% of the time that it's Tuesday will be because of Tails.

[bigmonkey]

Sorry for the delay. I'm going to try to find time (about 4 hours?) to read and respond to these tomorrow. Feel free to continue the debate though...

[Deorum]

[ QUOTE ]
[ QUOTE ]
"What is the probability that you were awakened by a tails-flip?" the answer is 2/3.
...
Since the amount of "heads flips" and "tails flips" are equivalent, she will state that the probability of the coin being flipped heads or tails is 50% for each.

[/ QUOTE ]

So, after being awakened, what is her answer to the question: "What is the probability that the coin was heads?"

The question is ambiguous. You give answers to both interpretations above.

[/ QUOTE ]

There is no ambiguity in the question. The answer to the question, "What is the probability that the coin was heads?" is 1/2. The answer to the question, "What is the probability that you were awakened this time by a 'tails flip'?" is 2/3.

These questions are not the same. The first refers to the probability of the state of the coin, while the second refers to the probability that she was awakened by a specific flip. The probability of the state of the coin and the probability that she was awakened by a specific flip are not the same.

Since different types of flips result in different numbers of awakenings, but flipping the coin results in an equal number of heads and tails, it follows that the probabilities for each question are different. However, neither question is ambiguous; both have definitive answers.

Edit: Check out the quick proof in my post above, it is pretty simplistic and will be the thought process which she would use to determine that the probability of either flip is 1/2. All of what is written above my quick proof is simply a more in depth explanation to the quick proof process.