1 point for each would also be a Dutch book wouldn't it? — Michael
Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, her credence that it is Monday should equal her credence that it is Tuesday since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus
P(Tails and Tuesday) = P(Tails and Monday).
Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. She knows the experimental procedure doesn't require the coin to actually be tossed until Tuesday morning, as the result only affects what happens after the Monday interview. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should therefore hold that P(Tails | Monday) = P(Heads | Monday), and thus
P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).
Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one-third by the previous two steps in the argument.
What's the probability that it's Monday? It's tempting to say 2/3, but that would be wrong. From the above, we know that P(Tails and Monday) = 0.25 and that P(Heads and Monday) = 0.5. So it must be that P(Monday) = 0.75. — Michael
This is Betting Game 2 from this post.My policy is to wager $1 on tails whenever I'm asked. You're paying even money.
If the toss is heads, I lose $1; this happens half the time, so my expected loss is $0.50.
If the toss is tails, I make $1 each time I'm asked; this scenario happens half the time, so I have an expected profit of $1.
So I make at least $0.50 on average each time I play, no matter how the toss goes.
I have made a Dutch book against you. — Srap Tasmaner
I can't quite follow this. I think there are a couple of grammar glitches in it that make it hard to understand what it is saying. Can you please expand on it, maybe using bullet points for clarity, and explain in what way you find it equivalent to the original question?Examiner tosses a fair coin, and then tosses another. If the first toss was tails, she asks Beauty her credence that the first toss came up heads; if the first toss was heads she only asks for Beauty's credence if the second toss was heads as well, otherwise the round is over.
Done this way, Beauty will know that when she was not asked the first toss was heads, but she can do nothing with that knowledge. She's not asked and the round is over. What matters is that she's always asked when it was tails and asked half the time when it was heads. So her credence that it was heads should be 1/3. — Srap Tasmaner
Ouch... Does miss Beauty know that she's getting fewer points on some of the bets. In that case the outcome is certain and she can win every bet. If not, she's not the one doing the gambling.So, I ran 100,000 games and gave 1 point for successfully guessing heads and 0.5 points for successfully guessing tails (because you get two opportunities). It doesn't matter if you always select heads, always select tails, select tails 1/2 the time, or select tails 2/3 of the time. The average score is 0.5 in every case. — Michael
This whole betting thing demonstrates the correct answer. It only needs to be done twice, not 100000 times, because there are only two unique cases occurring in equal probability.
I wake up and am expected to bet a coin with even odds. I bet tails and win 2 coins if it is tails, and lose one coin if it is heads. Sounds like a winning bet to me. — noAxioms
I am not convinced it contradicts the 1/3, but instead just leads back to the initial problem of the inherited subjectivity in the notion that you should start with a prior belief and update that prior into a posterior belief when you get new information. This carries that subjectivity into any posterior, and as we see here those priors can look very different based on the viewer. Not really objective science. — Jeremiah
Any takers on how to define a probability space in which the statement 'today is Monday' corresponds to a well-defined Event? — andrewk
And in what way is that not happening twice to her? She gets to bet twice in that case, despite the fact that she is unaware of which times she's betting twice.It doesn't happen twice to her. It only happens once to her, given that it was only tossed once. She just wakes up to it twice. — Michael
This is Betting Game 2 from this post. — andrewk
Game 2: At each interview, Beauty bets $1 to guess what coin came up, and loses that dollar if wrong or wins $2 if right. — andrewk
My calculation is that expecting that profit is the same as if there were a single $1 bet at even odds that the result of a coin toss will be tails and the coin had a 3/4 probability of coming up tails. To see this, note that in that case the expected profit is
$1 x 3/4 + (-$1) x 1/4 = $0.50
Hence, in interpreting the term 'degree of belief' it seems reasonable to make it the probability of tails when betting on a single flip of an unfair coin, at which one would have the same expected profit. So under this interpretation of 'degree of belief', the answer is 3/4 for tails, and hence 1/4 for heads. — andrewk
Can you please expand on it — andrewk
Game 2: At each interview, Beauty bets $1 to guess what coin came up, and loses that dollar if wrong or wins $2 if right — andrewk
I don't know betting terminology, so I may have used the wrong words. What I meant by 'wins $2' is that she gets her own dollar back, plus another dollar. Perhaps the correct betting terminology for that is 'wins $1'. My ignorance of gambling terminology is gargantuan.This is why I said I couldn't understand how you were setting the odds. Doing it this way is paying off 2-1 on both heads and tails, which is incoherent. (Unless you meant both tails interviews would taken together pay off $2, but that's still incoherent.) — Srap Tasmaner
Yes I think we can agree that if a bookmaker offers Beauty Game 2, which has even odds, Beauty can select a strategy under which the expected value of the bookmaker's profit is negative.Either way -- 2-1 or 3-1 -- offering even odds in this game loses money. We agree on that, right?
So put your money where your mouth is. What should you bet? I say bet £1 on heads. — Michael
Sure, but it doesn't mean that tails is twice as likely to occur as heads, which is why these betting examples miss the point. All the betting examples show is that it's better to bet on whichever outcome provides more payouts, which is obvious. — Michael
What I'm saying is that there is no reason for her to have a greater belief that it was tails than heads. When she's asked what her belief is that it was heads the rational answer is 50:50. — Michael
in interpreting the term 'degree of belief' it seems reasonable to make it the probability of tails when betting on a single flip of an unfair coin — andrewk
Assume she's woken on Monday if it's heads or Tuesday and Wednesday if it's tails.
Do you agree that P(Monday|Awake) = 1/2? — Michael
Mon Tue Heads 1/2 0 Tails 1/4 1/4
Mon Tue Heads 1/3 0 Tails 1/3 1/3
Except in this, and in the Monty Hall problem, there is no new information. — tom
I don't understand at all what you mean by P(Monday|Heads) and P(Monday|Tails). — Srap Tasmaner
And in what way is that not happening twice to her? She gets to bet twice in that case, despite the fact that she is unaware of which times she's betting twice. — noAxioms
Thirder:
Mon Tue
Heads 1/3 0
Tails 1/3 1/3
P(Heads|Awake) = 1/3
P(Heads|Monday) = P(Heads and Monday) / P(Monday) = 1/3 / 2/3 = 1/2
One characteristic of the thirder view is that it doesn't imply an external perspective that "knows" what the probability is for Beauty's current awake state. Since Beauty has no available information distinguishing the three states from her point-of-view, she is simply indifferent about which state she is currently in, and so assigns a probability of 1/3 for each awake state.
So the thirder view can be preferred on pragmatic grounds. — Andrew M
Mon Tue Roll 1-5 1/3 0 Roll 6 1/3 1/3
Mon Tue Roll 1-5 5/6 0 Roll 6 1/12 1/12
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. The interview consists of her being asked whether the coin toss was heads or tails, if she guesses wrong, she'll be executed. If she still lives, she will be awakened on Wednesday without interview and the experiment ends.
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