This is a fallacy:
If Monday, P(Monday-Heads) = P(Monday-Tails)
If Tails, P(Monday-Tails) = P(Tuesday-Tails)
Therefore, P(Monday-Heads) = P(Monday-Tails) = P(Tuesday-Tails)
The conclusion doesn't follow, because the first two equalities depend on the conditionals being true.
You can see this by observing that
P(Monday-Heads) = 1/2
P(Monday-Tails) = 1/4
P(Tuesday-Tails) = 1/4
Also satisfies the two conditional statements, without satisfying the conclusion — hypericin
Your claim that "The next enclosure is the toucan enclosure iff I first turned right at the fork (P = 1/2) and then passed the tiger enclosure," is an assumption that can't be put forward without begging the question against the Thirder. You need to substantiate, rather than presuppose, that when you're nearing an enclosure, there's a 1/2 chance the path you're on is a T-path. — Pierre-Normand
It's irrelevant (it refers to occurrences after she has answered). But I did intend to take that one out.It's not. You say:
"If the coin landed on Heads, then an N-sided die is rolled, where N>=2. She is woken on day D1 - that is, D1 days after day 0 - where D1 is the result of this roll, and asked her credence. Then she is put back to sleep." — Michael
In my example she isn't put back to sleep. The experiment just ends. The same with her second tails interview. So we have no idea how many days the experiment will last. It could be anywhere between 1 and N days.
On each day in the range 1 to N, the prior probability that she will be woken on that day AND the coin landed on Heads is 1/N. — JeffJo
Fixed; I was in a hurry, and that didn't affect the answer. All the probabilities I gave were off by that factor of 1/2.Pr(Heads & Day = D) = 1/2 * 1/N. — Michael
No, the prior probability that she will be woken on Tuesday, and the coin landed Heads, is 1/4. The prior probability that she is awake and the coin landed Heads is 0. "Will be woken" and "is awake" are not the same events.That aside, using your above reasoning, in the normal problem the prior probability that she will be woken on Tuesday and the coin landed on Heads is 0
And for about the tenth time, "rules out" is not a valid expression. I only use it since you can't stop using it, and then only when I really mean a valid one. The conditional probability of event A, given event C, is defined to be:So when she "rules out" Pr(Heads & Day = Tuesday)
No, the prior probability that she will be woken on Tuesday, and the coin landed Heads, is 1/4. — JeffJo
Both of these are true (note the tense):
1. To reach the toucon enclosure I must first turn right at the fork and then pass the tiger enclosure
2. The probability that I will turn right at the fork is 1/2
When I wake and consider my credence that the next enclosure is the toucon enclosure I consider what must have happened (or not happened) for the next enclosure to be the toucon enclosure. I know that I must have first turned right at the fork (A) and then passed the tiger enclosure (B).
P(A, B) = P(A) × P(B|A)
My claim is that the probability of having turned right at the fork is equal to the probability of turning right at the fork, i.e. 1/2.
Your claim is that the probability of having turned right at the fork is equal to the fraction of all encountered enclosures which are right-side enclosures, i.e. 2/3.
I don't think your claim makes any sense. The probability of the first event having happened isn't determined by what could happen after that first event happens. The probability of the first event having happened is determined only by the probability of that first event happening. — Michael
If a die rolls a 6 then Sleeping Beauty is woken six times otherwise she is woken once. When woken what is her credence that the die rolled a 6?
Halfers have to say 1/6
and thirders have to say 6/11.
Before she is first put to sleep she is to bet on whether or not the die will roll a 6 – paid out at the end of the experiment – and each time she is woken she is allowed to change her bet.
If she bets according to her credence then both halfers and thirders have to say that before she is first put to sleep she will bet that the die will not roll a 6.
Thirders then have to say that when woken she will change her bet and bet that the die did roll a 6.
Are thirders willing to commit to their position and change their bet? — Michael
Beauty gets cloned with all her memories on a given flip, such that each Monday and Tuesday has a 50% chance of resulting in a new clone being created. — Count Timothy von Icarus
Based on (an admittedly simple) Bayesian take, Beauty should be increasingly confident that she is the real Beauty with each passing week. The whole idea is that repeated trials should move the dial in our probability estimates. And yet, this doesn't seem right, no?
Therefore, the "bet" one ought to make doesn't straightforwardly track one's credence in the outcome of the die roll, but rather, it must take into account the rules of payout in this specific experimental setup. — Pierre-Normand
They ask her one question after each time she awakens, however: What is the probability that the coin shows heads?
If each outcome has the same reward then it is rational to bet on the most probable outcome.
Therefore, if her credence that the die landed on a 6 is 6/11
6
11
then she will change her bet. Therefore, if she doesn't change her bet then her credence that the die landed on a 6 isn't 6/11
6
11
. — Michael
Sorry to resurrect. — JeffJo
The reason why SB can take a thirder rather than a halfer stance regarding her current awakening episode is because she may care about the long-term average frequency of such events (6-awakenings) — Pierre-Normand
She isn't being asked "what is the long-term average frequency of being woken up when the die did land on a 6?" — Michael
This is your favored interpretation. — Pierre-Normand
My "favoured" interpretation is the literal interpretation; she is being asked about the probability that a die rolled a six. — Michael
The problem only exists when the question being answered before being put sleep is the same question being answered after being woken up, and where the answer changes despite no new information.
If the Thirder's answer before being put to sleep is 1/6 and if their answer after being put to sleep is 6/11 then either they are not answering the same question or their answer is wrong.
Thirders then claim that:
P(6|Monday)=6/11
P(¬6|Monday)=5/11 — Michael
However, owing to the fact that the traveller must establish their credence on the occasion of encountering one among a set of indistinguishable doors, and 2/3rds of such doors belong to two-door dwellings, their credence that this house that they now are facing is a two-door dwelling is 2/3. — Pierre-Normand
Sorry, I deleted that post because it's late and I'm tired and I may have messed up the specific numbers. The general gist is what I said before. Your argument is that her reasoning after being woken up is:
A1. If I keep my bet and the die didn't land on a 6 then I will win £100
A2. If I change my bet and the die did land on a 6 then I will win £100
A3. My credence that the die landed on a 6 is 6/11
A4. Therefore, the expected return if I keep my bet is £83.33
A5. Therefore, the expected return if I change my bet is £16.67
But A3, A4, and A5 are inconsistent. If A3 really was true then she would calculate different values for A4 and A5, concluding that it is profitable to change her bet. But she doesn't do this. — Michael
A thirder will not agree with A4 or A5. — Pierre-Normand
All this shows is that the lopsided payout structure makes it irrational for her to bet on the most likely outcome. — Pierre-Normand
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