Yes, there is something absurd about the 2/3, but it's a result of putting in twice as many blues per toss but then taking them out one at a time, as if they were the same as the reds. — Srap Tasmaner
Mon Tue Wed ... Heads 1/2 Tails 1/2000 1/2000 1/2000 ...
The thing is, this 2:1 proportion of interviews is right, but remember that SB does not payout like a wager on a 2:1 biased coin. It pays out like a 3:1 coin. — Srap Tasmaner
Yes, this is an important point. I think the intuitive comparison with a weighted coin is misleading since SB is just structured differently. Adding more interviews (and thus bets) on tails is not like increasing coin bias. — Andrew M
If you’re told it’s Monday then the probability is 1/2 that it’s heads and if you’re told it’s Tuesday then the probability is 0. — Michael
I'd like to analyze the halfer's P(Heads|Monday) = 2/3 consequence further — Andrew M
That extra blue marble left in the hopper is not another outcome; it's just the rest of the outcome you already know about from the first blue marble. — Srap Tasmaner
This position is attractive, but I just don't understand how it works. — Srap Tasmaner
This is worked out not by some equation but just by knowing the rules and how coin flips work. — Michael
The thirder reasoning works against intuition as well, especially in the extreme version. It suggests that if we’re to be woken a thousand times in the case of tails then we should be almost certain that it’s tails upon waking, despite the fact that it’s an unbiased coin flip. And all because if we’re right then we’re right more often? That shouldn’t be the measure. — Michael
Indeed, I think it means that the odds here are not truly 2:1 at all. — Srap Tasmaner
I can't figure out how to make this into a normal wager of any kind. — Srap Tasmaner
In the Sleeping Beauty scenario, those two methods give different answers. The first method is the halfer view which understands the coin to be in one of two states. But that leads to absurdity when conditioning on Monday. — Andrew M
I think the halfer reasoning should just be that it’s a 50:50 chance that it’s heads, whether unconditioned or conditioned to Monday. — Michael
Mon Tue H 1/4 1/4 T 1/4 1/4
Mon Tue H 1/2 0 T 1/4 1/4
I think probability is a measure of self-locating uncertainty in a state space. It's not directly about coin outcomes, days, or even interviews at all, except in so far as they contribute to the construction of the state space. — Andrew M
That's saying P(A | B) = P(A), and therefore A and B are independent events. Of course in one sense the coin toss and the day of the week are independent, but whether Beauty is interviewed on that day is clearly not independent of the coin toss — Srap Tasmaner
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.
So Monday is independent of the coin toss but ¬Monday isn't? — Srap Tasmaner
Speaking of ambiguity, does when the coin toss happens affect Beauty's credences? Don't we want a solution that applies to a toss before the Monday interview as well as after? — Srap Tasmaner
I think the halfer reasoning should just be that it’s a 50:50 chance that it’s heads, whether unconditioned or conditioned to Monday. We shouldn’t be applying some formula and should just consider what we know about coin flips. — Michael
P(Heads) = 1/2 P(Heads|Monday) = 1/2 Mon Tue Mon Tue Heads 1/2 Heads 1/2 Tails 1/4 1/4 Tails 1/2
The thirder model relies explicitly on there being a single toss of a coin with heads and tails distributed 50:50. (And we've agreed you cannot construct an alternate model with a weighted coin.) How can Beauty take that as a premise and then be unable to reach the conclusion that the chances of heads were 1/2? — Srap Tasmaner
You also accept that conditionalization changes Beauty's probability of heads when she is told it is Monday — Andrew M
Mon Tue Total Heads Awake:$1 Asleep:$2 $3 Tails Awake:$4 Awake: $8 $12
I think you are both looking at the experiment from an independent observer's perspective (or Beauty's Sunday perspective) and not from Beauty's perspective when she is awakened and interviewed in the experiment. — Andrew M
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