The betting matters, and be honest: you were happy enough to use wagering arguments when it suited you. (What's more the examples don't show what you say is obvious; they assume it and it's irrelevant here anyway.) — Srap Tasmaner
Or a weighted coin that’s 5/6 chance of heads. — Michael
A simple, discrete probability space consists of two things - a sample space, which is the set of all possible outcomes, called Events, and probabilities of each Event.Is that what I've done above?
P(Tails and Monday) = P(Tails) * P(Monday|Tails) = 0.5 * 0.5 = 0.25
P(Heads and Monday) = P(Heads) * P(Monday|Heads) = 0.5 * 1 = 0.5
Therefore, P(Monday) = 0.75 — Michael
That is, five states that come up heads and one state that comes up tails. — Andrew M
I don't feel that this changes the situation, because although there are two events at which saying Heads can get Beauty killed, if she decides to say Heads, she will never make it to the second event, so the event of being killed at the second waking has a zero probability of occurring.I liked the solution I read on the physics forum from PeroK. Let's change the story shall we?
*** 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. ***
Given these facts, what answer will Sleeping Beauty give?
If the toss is heads, she will awaken Monday. If she'd then say heads, she lives. If she'd say tails she'd be dead.
If the toss is tails, she will awaken first on Monday. If she then says heads, she dies. If she says tails, she'll live.
If she said tails, she will awaken again on Tuesday. If she then says heads, she dies. If she says tails, she'll live.
In the above we see there's only one event where saying tails gets you killed and two events where saying heads gets you killed. Sleeping Beauty would be smartest to state tails. — Benkei
I am inclined to conclude that, if we try to use probabilities rather than betting to solve this, we are left with a degree of freedom - the value of p - that cannot be removed without making a controversial assertion about how Beauty should assign probabilities in her epistemological probability space. — andrewk
I don't feel that this changes the situation, because although there are two events at which saying Heads can get Beauty killed, if she decides to say Heads, she will never make it to the second event, so the event of being killed at the second waking has a zero probability of occurring. — andrewk
I don't see how answering heads would be an equally safe bet. — Benkei
It wouldn't be a safe bet, but that's irrelevant.
If I offer you one free lottery ticket if you correctly guess heads and two free lottery tickets if you correctly guess tails then tails is the better bet even though equally likely. — Michael
Even if she knows there's a 50% chance of either, the fact that it could already be Tuesday would change my assessment if I were in her shoes. — Benkei
You say it is 50% odds, but tails is the better bet. This seems contradictory to me.That doesn't mean it's more likely to be tails. It just means that tails is the better bet. — Michael
You say it is 50% odds, but tails is the better bet. This seems contradictory to me. — noAxioms
The odds of the flip are 50% from nobody's point of view. They seems to be 33% (the one point where neither is the better bet) from Beauty's POV, and they are 100% from everybody else's POV since they know the outcome of the flip during any of the wakings. It is 50% only from the external POV (not Beauty) only before the toss, which is not during any of the wakings. — noAxioms
The probability that it's Monday given the fact that it's heads/tails — Michael
This is not the situation faced by Beauty. She is offered one bet that wins or loses one ticket. There is no double payout during any of here wakings.If I offer you one free lottery ticket if you correctly guess heads and two free lottery tickets if you correctly guess tails then tails is the better bet even though equally likely.
Tails isn't the better bet because it's more likely but because it has a better payout. — Michael
Yes, it is about credence. Beauty has information about the coin toss, and that alters the credence from the 50/50 credence that exists to nobody in the scenario.It's not. It's about her credence. — Michael
Yes, because I've been given no more information, so the odds remain 50%.If you tell me that you flipped a coin ten minutes ago I'm going to say that there's a 50% chance that it landed heads. — Michael
This is a different scenario. When do I find out that I get a second guess? You would have to either tell me before the first guess, or after it. If after, odds are 50% on the first bet and 100% on the 2nd. If before, then 100% on both.And if you tell me that I'll get £1 for guessing correctly, and that you'll let me guess twice if it's tails, — Michael
In the Monty Hall problem, the host gives you information that changes the probabilities that you assign to each door. That information is new to you.
Similarly, in the Sleeping Beauty problem, awakening provides information that enables you to rule out one of the four states. However since you have no information distinguishing the remaining states, you should be indifferent about which state you are currently in. — Andrew M
'will happen' implies the coin has not yet been tossed. It has, and today has happened, which yields information that changes the credence. That changes the odds of B and D to 66%, 33% each.There's a 50% chance that both B and D will happen, but there's a 25% chance that today is B. — Michael
That changes the odds of B and D to 66%, 33% each. — noAxioms
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