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Okay, but what if Bob is a Christian baby…?
Maybe it’s the fact that it’s the end of a long day, or because I dealt with an AI-psychotic crackpot earlier, but I can’t follow this at all. Surely there’s a more elegant framing.
I definitely don’t see why it’s culture war. Not unless this is a devious way to criticize the woke left.
The culture war angle is that the correct answer is 1/3 and that the people who think it is 1/2 cannot comprehend a word problem and need to be put into UCSD's remedial mathematics course.
Can't tell if really good joke because that's what we actually see the culture warriors roll with... or if actually missed the point.
I phrased it a bit flippantly, but I do think that the original question is not ambiguously phrased. We do not say that people who think that there is no advantage to switching doors in the Monty Hall problem are answering a different question than the people who say that there is an advantage to switching. We say they are wrong.
Verily, in the Monty Hall problem. There, you actually do have a very very clear moment where information is gained and there is no ambiguity about which question you are being asked. But in this problem, if Alice tells Bob what you seem to want to have her tell him, we would say that she is wrong. We'd even say that she's extra wrong if she said she "updated".
Sorry, my comment was ambiguously phrased. I was referring to the cannonical form of the Sleeping Beauty question from Wikipedia:
This question is not ambiguous. The correct answer is 1/3. If you ran this experiment on people who think the answer is 1/2 you could take their money.
I'll bite. Assume I believe the answer is 1/2, how would you take my money?
You are Sleeping Beauty, I am the Magician. The experimental setup is exactly the same as outlined in my previous comment, except that you deposit $20,000 with me before the experiment starts. I explain that each time I wake you up, I will hand you $10,000 of your own money and give you the option to bet at 3:2 odds that the coin came up heads. At the end of the experiment, any unbet deposits will be returned to you.
When you wake up, if you think that there is a 50% chance that the coin came up heads, then you should bet the $10,000 (because 3:2 is a better payout than 1:1). You have no way of determining which situation you woke up into, so you should take the bet every time if this is your true belief.
I am thrilled to offer you this bet. From my perspective, there is a 50% chance that the coin comes up heads, in which case you win $15,000 from me. However, the other 50% chance is that the coin came up tails, in which case you woke up and bet $10,000 on heads twice, so I won $20,000 from you. The net outcome is:
Ah but in this construction, I have to pay my stake twice. That's not actually a 3:2 bet, but a 3:4 bet. I'm staking my $20,000 against your $15,000 after all.
Recall that the original formulation was:
If similarly, the bet only happens during the first awakening, then clearly it is in my favor to take the bet, since it's equally likely that the coin came up heads or tails.
If we're betting on every awakening, that means that if I lose on timeline B, I have to pay twice. So I'm not actually betting on 3:2 odds, but on 3:4 odds, which are clearly not in my favor if I believe the odds are 1:1, and I wouldn't take that bet.
I read up a bit on the older thread and I liked the analogy about putting balls in a bag. If the coin toss comes up head, then we put 1 red ball in the bag, if it comes up tail, we put 2 green balls in the bag. Clearly all the balls in the bag are either red or green.
If you ask me: “what is the probability that the balls in the bag are red?” I would say 50%.
If you ask me: “do you take a bet where I pay you $15 if the balls in the bag are red, and you pay me $10 if the balls in the bag are green?” I would take that bet since I know the odds are even and so the expected value is positive ($2.5).
If you ask me: “do you take a bet where I pay you $15 for each red ball in the bag, and you pay me $10 for each green ball in the bag?” I would not take that bet because I know that the probability of the balls being red/green are 1:1, but if they are green, there are 2 of them, so you'll pay me $15 in one case, and I pay you $20 in the other, which is a negative expected value (-$2.5).
So when first awakened Sleeping Beauty ought to believe that the coin came up heads with 50% probability. She also oughtn't take a bet that is resolved on every awakening, knowing that awakenings happen more frequently in timeline B. If she is guaranteed her bet is only resolved on the first awakening, she can take the bet.
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