The Sleeping Beauty paradox

Great Debates
101

Who says I’ll be right 1000/1001 times if I bet my own gender?

Day 1: God flips the coin, it comes up heads. He creates me as the sole man on Earth. I bet my own gender dominates; I am wrong. God decides to destroy the universe and start over.
Day 2: God flips the coin, it comes up heads. He creates me as one among 1000 men. I bet my own gender dominates; I am right. Destroy, restart.
Day 3: Repeat of Day. Day 4: Repeat of Day 2. Etc. Half the days, the coin comes up heads, and half the days, it comes up tails, as would be expected from a fair coin.

I bet my own gender dominates, but I’m right only 50% of the time, not 1000/1001 times, in this situation.

OR

Day 1: God flips the coin, it comes up heads. He creates me as the sole man on Earth. I bet my own gender dominates; I am wrong.
Day 2: God flips the coin, it comes up tails. He creates me as the sole women on Earth. I bet my own gender dominates; I am wrong.
Day 3: Repeat Day 1. Day 4: Repeat Day 2. Etc. Half the time, the coin comes up heads. But all the time, I’m wrong.

OR

Day 1: Coin comes up heads, I am created as sole man. Bet my own gender and am wrong.
Day 2: Tails, I am sole woman. Bet own gender and am wrong.
Day 3: Heads, but I’m not even created this time, just a bunch of other men and women. I don’t even get a chance to bet.
Day 4: Tails, I am one among many women. Bet my own gender and am right.
etc., in some complicated pattern, such that, in the end, even though the coin comes up heads half the time and tails half the time, I only get to bet 45% of the time, and out of those times, I’m only correct 13% of the time.

What says it doesn’t work like any of these instead? In light of these, whence the conclusion “If you bet your own gender dominates, you’ll be right 1000/1001 times”?

102

Situations like that are pretty standard in statistics. Everyone computes statistical tests to some level of confidence (maybe 90% or 95%), so even in the best scenario, one out of 20 statistical conclusions is wrong. Since everyone was guaranteed to meet someone of the opposing view, you can’t get any new information out of the situation, and so everyone should remain with their original conclusion. After all, they still have 1000/1001 chance of being correct.

Indistinguishable, regarding your latest responce, I came up with something to address exactly that point, and decided against posting it. It’s the end of the workday here, so I’m going to head home. I think I should be able to get a responce up later this evening.

103

This is like if Monty Hall shows you a 1001 doors, one of which you randomly select to open, revealing a man. Then Monty Hall refers to his chart of room occupants and deliberately selects another door, revealing a woman. The fact that Monty can find a woman is meaningless - he has the room occupant chart to refer to. Because he hasn’t given you any more meaningful information, you should not change your opinion.

Amusingly, neither should the woman - from her perspective, she is a random one of the thousand, and you are a non-random selection matched to her. Of course, one of you is wrong - but we always knew that one man, or one woman, would be wrong. All that you know by meeting as described is that that one wrong person is in the room - which doesn’t hint at which one of you is that person.
Now, if you were allowed to open up another door at random and then found a woman there, then you should waver - the random chances would balance out. And if you opened up another door and found a woman, well, we’d be done :smiley: -but if there were two of one gender and a two thousand of the other gender, finding the second random woman should make you nigh-certain that the rest are all women too.

104

My way of looking it would be this… The paradox is logically equivalent to doing this…

Take a bowl and bunch of red/blue balls…
Toss a coin
If its heads put a single red ball into bowl
If its tails put two blue balls into bowl
What is the chance the coin toss was heads ? - 1/2
What is the chance a random ball is red ? - 1/3

No paradox there, its just two different questions. There is no “point of view” involved.

105

Case three is invalid - only people who are created count. Imaginary uncreated people don’t - which is good, because there are an infinite amount of imaginary uncreated people. (If you’re trying to analogize not-created to not-being-asked, you’re mixing up the analogy; one of the genders represents not-being-asked, give or take.)
And in the first two cases, you’re ignoring that you’re not the only person on the planet. Everyone gets a vote, and if everyone bets their own gender, then they’re not (as an aggregate) wrong half the time, 1/1001th of the population is wrong, each and every time. The half part comes in the fact that half the time, that wrong person is a woman.

106

Missed the edit window - this bit I said here is completely wrong. :smack: The genders analogize the two different possible sequences of days; so in the original scenario only one gender/set of interrogations would actually exist, when the dust settled. Of course, with probability we also consider the cases that don’t exist (though rarely do they get to meet and chat!), but that doesn’t excuse my cognitive flatulence here.

If anything, the non-existent/non-conscious people analogize the days after the experiment is over, when no questions are being asked. Of course, in reality there are exactly as many total days one way or the other - so the question become not one of how many people/days there are, but rather what are the odds that you, being a random conscious person, are among the set with 1000 conscious people , as opposed to being in the set with 1 conscious person.

107

To the replies to my post #100:
So even though the communicating man and the woman have the exact same information at their disposal (they know the exact same facts about the universe), they should come to different conclusions?

108

Well, no, they know slightly different information. John knows that the creator would show him a woman if one existed, but he happened to be born a male; Joan knows that the creator would show him a man if one existed, but she happened to be born a female. Important distinction.

109

John also knows that the creator would show Joan a woman if one existed, but she happened to be born a female. Joan also knows that the creator would show John a man if one existed, but he happened to be born a male.

They both know the same facts; after all, they’re in communication, so whatever one knows, the other finds out. Whatever empirical evidence the one has gathered, the other knows about as well. It’s as if they’ve both published papers on their findings and now have access to the same corpus of knowledge.

110

Actually, although I quoted and re-used this line, looking back on it and trying to see what it means, it’s a weird thing to say, and I’m not sure what you were getting at with it; are you sure you understand the setup correctly? No matter what, at least one woman will exist, and no matter what, people are shown someone from the opposite gender. And John is shown a woman. Precisely because of (not in spite of) the fact that he was born a male.

111

That’s cool what you’re doing there. You’re trying to get freaked out by the idea that the act of observing changes what truth is. I’m not sure what to tell you. Maybe you’re not getting how trivial all this is. Maybe you’re onto something. Observation-changing-truth is basically the theme of quantum mechanics. And none of that stuff sounds anything as trivial as this.

112

Hey! That’s just like special relativity!

113

In what respect? Special relativity implies that people with the same information should draw different conclusions?

What does “observation-changing-truth” have to do with anything in my posts?

(As as side note, I would not say “observation-changing-truth” is basically the theme of quantum mechanics, so much as the theme of pop culture bastardizations of quantum mechanics; I would think non-commuting operators and phased probability amplitudes (to be interpreted via the Born rule) are the theme of quantum mechanics. But I’m not really in a position to say…)

114

Well, this is rarely explained properly when dealing with time dilation, but that is the theme of the “twin paradox.” There is no such thing as “one twin went faster than the other and didn’t age.” Either twin can be said to be traveling faster than the other, and each sees that it is the other who is aging slowly! I found it similar to how each gender is certain that it is the other that’s infrequent.

Sure. Anyway, the act of observing is a pivotal thing in QM and underlies everything from the double-slit experiment to quantum computing. Incidentally, QM, like the SB riddle, is largely a study in probabilities.

Btw, I took this as the basis of my poetically-worded ‘observation-changing-truth’ comment, since I equate “reaching conclusions” to “figuring out the truth.” I mean… they’re pretty much the same phrases as far as colloquial English is concerned, right?

115

Indistinguishable: Like in all math, we have to make some assumptions in probability for the problems to be tractable. In the context of this problem, the natural assumption to make is the egalitarian one, that everyone is basically the same–that every potential person has an equal chance of filling any body. With this in mind, are your scenarios in post #101 meant to be random happenings, or examples of God screwing with you? If it’s the former, and we’re keeping our egalitarian assumption, then it’s possible to show mathematically that any infinite chain of events that does not have you waking up as a member of the majority 1000/1001 times has probability zero. If you mean the latter, then we’re outside of the scope of probability theory. If we’re going to drop the egalitarian assumption, then we need a reason to adopt another perspective.

Back to post #100. If we’re keeping the egalitarian assumption, the man and the woman do not have the same information–they each know the gender of they body they drew in the random lottery. The fact that they meet at least one person of the other gender is deterministic, so it cannot affect our probabilistic beliefs. That people who draw different random outputs come to different conclusions is a basic fact of statistics. The fact that is covered up is that everyone believes they are right with certainty 1000/1001, and in fact 1000 out of 1001 people are right, so the probabilistic analysis is correct.

116

But, I assume, the situations you are referring to are ones where people draw different random outputs and then never see the others’. People who draw different random outputs but then afterwards share their data generally end up drawing the same conclusions, since they end up conditioning their priors on the same combined data… or do you have a basic statistical counterexample to this principle?

Now, you could say, John and Joan don’t end up having the same combined data. John’s data is “I am male and you are female” while Joan’s data is “I am female and you are male.” But do we want to say that this kind of switch of deixis in phrasing a proposition (simply switching which referent gets called “I” and which gets called “you”) can substantively switch its implications?

117

Under normal circumstances, yes, they should combine their data and reach a new conclusion. But in the hypothetical, the data that they get from sharing is entirely determined by the data they originally had. You meet someone of the opposite sex with probability 1 regardless of whether there are 1000 people of your sex or the opposite, so you can’t use the fact that you met someone of the opposite sex to support either conclusion. That’s why everyone comes out with the same probability estimate as they had going in.

118

I’m sure that if I remembered I’d based my destination decision on a coin toss, I’d still feel that there is equal likelihood of London or Paris (50/50).

The same if I were sleeping beauty - 1/2.

I can get the 1/3 answer, oversampling of one outcome, but I wouldn’t think like that IRL.

119

Absolutely. Which, I am saying, perhaps suggests that they should both have the same probability estimate going in (I am reaching this potential conclusion by combining your specific observation that, in this case, each person’s prob going in = their prob going out with the plausible general principle that two people’s prob going out should be equal after they’ve shared data).

120

Actually, I’d say the difference in data they had was “I am one person, as likely to be one of the males in this group as anyone else in this group is, and I turned out to be male. You are a female and were specifically chosen as female because I turned out to be male; whether there was one female or a thousand has no impact on the gender you turned out to be, because it was predetermined by the gender I happened to turn out to be.”

The female has the reverse of that information; she knows that she is one individual out of a thousand and one, and she knows that the man she is presented with was selected from a smaller group that only included his gender.

When they meet, neither can see the other’s information, because they each know that that the other person was selected specifically for them, based on their gender, whereas they know that their own gender was not selected from a similarly filtered set. Quite simply the perspective each has on their own existence assures that the information remains non-transferrable.

Really this is very much like the usual Monty Hall question; there like here learning of the existence of something you already know (that there is at least one person of the opposite gender/that at least one of the other doors is wrong) tells you nothing and doesn’t change the odds of your original estimate being correct. (Of course, there the fact that one of the doors was removed from the set which collectively retains its 2:1 chance of being correct is helpful; here nothing is learned that changes anything.)