You forgot Heads and Wednesday, Heads and Thursday, Heads and Friday. . . . . and on forever.
Then do the same with Tails. — Jeremiah
And in the process don't forget the terms on the days Beauty will be awakened AND interviewed (AKA the sample space), was defined before the experiment started. — Jeremiah
. But since the probabilities are all equal, it doesn't matter how large the background sample space is (at least in a finite universe). — Andrew M
Oh really? So you think they are the same thing. — Jeremiah
Mon Tue Heads 1/3 Tails 1/3 1/3
H has one element and set T has two elements, each with a 50% chance of being selected. — Jeremiah
In the event of Tails, Beauty will be awakened on Monday and Tuesday, but due to the nature of the experiment she will not be able to tell the difference, either one is equally likely when interviewed — Jeremiah
So what? It's not a situation that arises. Neither she not the experimenters are ever in the position of knowing that the coin landed tails but wondering what day it is. Beauty only wonders what day it is to figure out how the coin landed. — Srap Tasmaner
The interviews are not randomly distributed. — Srap Tasmaner
The conditional probability of tails given that it is M2 or Tu is P(T|M2)= P(.5|.25) = .125/.25 = 1/2 = P(T|Tu) which is equal to P(H). — Jeremiah
P(H | M1) = 1, right? And this is the thing about the double interview track: both them happen if and only if the coin lands tails. From your calculation, P(T | M2 v Tu) = 1, yes? But it should be P(T | M2) = P(T | Tu) = 1, and P(T) = P(M2) = P(Tu) = 1/2. You always get both on tails. You get them one at a time, but we don't necessarily care.
That space of three possibilities, {M1, M2, Tu} has three elements each of which has an unconditional probability of 50%. Conditioned on the whole space, they'll each be 33%. — Srap Tasmaner
each blue is discounted, and is only half the total available evidence of a tails flip, unlike the reds each of which is all the evidence of a heads — Srap Tasmaner
A randomly selected marble is now twice as likely to be blue — Srap Tasmaner
but each blue is discounted, and is only half the total available evidence of a tails flip, unlike the reds each of which is all the evidence of a heads. — Srap Tasmaner
Discounting doesn't help Beauty. Suppose she is a halfer. Should she discount the next drawn marble (or current interview) by 1/2 or not? Well, she should 2/3 of the time. So 2/3 * (1/2 * 1/2) + 1/3 * 1/2 = 1/3. The thirder says that is the real probability of each state for her. — Andrew M
Mon Tue H 1/2 T 1/4 1/4
What are your odds of getting a red marble? 1/2. — Srap Tasmaner
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