Perhaps quote the axiom in question. I suspect it applies to something not known, such as a coin toss that has not yet been made, or which has been put under a cup without observation. Beauty has been given information about the toss, and that cannot leave the odds unaltered. She has been informed that the combination of Tuesday and Heads is not the case. That information could not have been conveyed if the coin toss was still under the cup. If she had full information ("it was tails"), then the odds would change to 100% tails, axiom of probability not withstanding.Why? If the flip had a 50% chance of being heads and if heads guarantees Monday then there's a 50% chance that today is heads and Monday. You seem to just be asserting these probabilities without adequately explaining how you got there. I get to my probabilities by applying an axiom of probability. — Michael
OK, I didn't know that's what that was. I'm actually not much up on the notation of it all, so I have a helluva time following it.I've done so multiple times:
Kolmogorov definition:
<sorry, the formatting rendered the quote unreadable> — Michael
I don't see how you are applying the additional information of 'it isn't Tuesday/heads' into your computation. — noAxioms
So yes, unconditional odds of heads is 50%, but Beauty is not working from unconditional — noAxioms
Respond then to my post about her getting to wake up on Tuesday.Heads as well. It spells that out.But it doesn't follow from that that each of the other three outcomes are equally likely. — Michael
I don't see how you are applying the additional information of 'it isn't Tuesday/heads' into your computation.
— noAxioms
That's the P(Monday|Heads) = 1. — Michael
Excuse me if I am new to the notation. I read this as the probability of it being at least one of Monday or Heads is 1, but since it might be Tuesday/Tails, this is wrong. I would think the probability of Monday or Tails is certain.
Maybe I just don't know how to read the notation. — noAxioms
As you see, the quote is getting altered. Sorry.So what's the condition? P(Heads|Awake)? Well, let's apply the Kolmogorov definition again:
P(Heads|Awake)=P(Heads∩Awake)/P(Awake)
P(Heads|Awake)=0.5/1=0.5 — Michael
Oh crap. OK then. I really don't know how to read this stuff then. '|' means 'or' in my world, but they have that intersection symbol to mean that here. Union for 'and'.P(Monday|Heads) means "the probability that it's Monday given the fact that it's heads" — Michael
Also since when did a Bayesian approach become the gold standard, what about a Classical approach? — Jeremiah
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. — Andrew M
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
If the odds were 1/2 and the Sleeping Beauty got to bet 1€ each time she was woken up, she should break even no matter what, but we notice that betting tails wins her 2€ with tails and loses her 1€ with heads. — BlueBanana
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
You're conflating "more likely to win if tails" and "more likely that tails". — Michael
But she's only given 1€ with tails. The reason she wins by guessing tails is because she's likelier to be in a situation where tails has been thrown. — BlueBanana
She's given it twice: once on Monday and once on Tuesday. — Michael
Because of the amnesia different people can be used to compare. Heads, I randomly choose a person from the street to ask the question from. Tails, I choose a hundred. Should they guess I threw tails or heads? — BlueBanana
For any given person there's a 50% chance that they're right, so it doesn't matter if they pick heads or tails. It's just that if it's tails and they pick heads then there's a greater number of losers and if it's tails and they picks tails then there's a greater number of winners. — Michael
But you don't say that if there's more winners under tails then tails is more likely. That's a non sequitur. — Michael
There aren't more winners because it's more likely but because you asked more people. — Michael
If they make the sensible choice of tails there're more winners because it's the likelier choice to be the correct one. — BlueBanana
It's more likely because I asked more people. — BlueBanana
There are more winners because you asked more people. — Michael
It doesn't make tails more likely to be the result of my throw, but it makes it likelier for the correct answer to be tails. — BlueBanana
All you're saying is:
If 100 people asked then it was tails [the rule of the game]
100 winners [the outcome]
Therefore, it was tails
But that's obvious, and not relevant. — Michael
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