Something “funny” happens exactly in step 4. Specifically: Instead of pairing you with a random person, the angel is pairing you with a person selected by specific criteria “if you rolled 6, the other person didn’t; and if you didn’t, the other person did”. Therefore, when meeting the other person, you should abandon the intuition that you both are a randomly selected pair.
The confusing part is that the situation is symetrical for both people. Yes, it is! But that is confusing only because we work with infinity here. You can rearrange an infinite set to increase or decrease a measure of its subset; and this is exactly what happens in step 4.
Step by thep: At the beginning, everyone’s (that includes you) chance to have 6 was 1⁄6. So there was 1⁄6 of people with sixes, and 5⁄6 of people without sixes. Then the angel rearranged people, so the measure of people with sixes was increased to 1⁄2, and the measure of people without sixes was decreased to 1⁄2. Now at the end, your probability remains 1⁄6, so the other person’s probability is 5⁄6. -- The confusing part is that the other person at the beginning also had chance 1⁄6 to have 6. But if they had 6 and you didn’t, then there is a higher probability that the angel will assign them to you.
In other words, you should believe that their chance is 5⁄6 in the sense they if they didn’t roll 6, they would be more likely to be assigned by the angel to someone who did; but instead they were assigned to you (and you with probability 5⁄6 are a person who didn’t roll 6).
It is important to remember that first we rolled dice, and only later the angel decided that we two will be a pair. If both of us would not roll 6, the angel would simply decide to not bring us together. -- The paradox of infinity is that the angel will always have enough other people to bring, if we both happen to not roll 6.
Let’s make it simple by removing the infinities.
There are exactly one million people, each of them rolls a die. An angel blindfolds them, and sorts them into two large groups: those who rolled a six, and those who didn’t. The blindfolded people cannot compare the sizes of their groups. Then the angel asks you: “What is the probability of you having rolled six?” (You can assume that the angel speaks simultaneously with everyone, so there is nothing special about asking you this question, instead of asking it someone else.)
A logical answer would be: 1⁄6.
Then the angel says: “Now I choose a random person from the other group; let’s call him Joe. What is the probability that Joe has rolled a six?”
A logical answer: 5⁄6.
The angel says: “However, Joe believes his chance of having rolled a six is 1⁄6, and he uses the same reasoning that you do. Is he wrong or what? Remember that I am having this dialogue simultaneously with each participant of the experiment.”
A logical answer: Probability is in the mind. In my mind, Joe has a probability 5⁄6 of having rolled a six. In Joe’s mind, he has a probability 1⁄6 of having rolled a six. These are two different questions.
Imagine that there are six people: me, Joe, Adam, Betty, Cecil, Daniel; and exactly one of us has rolled a 6. If it was me who rolled the six, you could have chosen any of {Joe, Adam, Betty, Cecil, Daniel} to be my twin. But if it was Joe who rolled the six, you didn’t have another choice for my twin, only Joe. So the fact that Joe is my twin, is for me an evidence that Joe has rolled the six. And symetrically, if you chose me as a twin for Joe, that is for him an evidence that I have rolled the six.
Because we have this finite scenario, one person was chosen five times as a “twin”, one person was chosen once, and four people we not chosen as “twins” at all. So we have only one of six people rolling 6, but we have five of six “twins” rolling 6. Therefore my probability of rolling 6 is 1⁄6 and my “twin”’s probability is 5⁄6.
...however in the infinite version, even if one subset is 5 times smaller than the other, the angel can choose a twin for everyone. But in some sense it means that if you rolled six, you are five times more likely to be selected as someone’s pair, as you would be if the angel just ignored the numbers on dice and paired people randomly.
In my mind, Joe has a probability 5⁄6 of having rolled a six. In Joe’s mind, he has a probability 1⁄6 of having rolled a six. These are two different questions.
While they are two different questions, you both have the same evidence and the same priors. They are symmetric questions, and should have identical answers.
[ETA]: It’s important to remember that the probabilities are relative; you have odds of 1⁄6, you should posit odds of 5⁄6 for your twin. Likewise, your twin has odds of 1⁄6, and your twin should posit odds of 5⁄6 for you.
Are you selling the outcome? I/e, “If I rolled a 6, you can have the $1”?
Because yes, both players should choose to sell the option. And likewise, both should choose to buy if their twin offers. Or, alternatively, both players should rationally agree to exchange outcomes.
And precisely because of how the angel assigned the twins, people would be winning on average in this game.
With infinities, this is possible. Just like in the infinite world, pyramid schemes are possible. You can generate free money simply by having each person with number N send $100 to a person with number N/2.
Something “funny” happens exactly in step 4. Specifically: Instead of pairing you with a random person, the angel is pairing you with a person selected by specific criteria “if you rolled 6, the other person didn’t; and if you didn’t, the other person did”. Therefore, when meeting the other person, you should abandon the intuition that you both are a randomly selected pair.
The confusing part is that the situation is symetrical for both people. Yes, it is! But that is confusing only because we work with infinity here. You can rearrange an infinite set to increase or decrease a measure of its subset; and this is exactly what happens in step 4.
Step by thep: At the beginning, everyone’s (that includes you) chance to have 6 was 1⁄6. So there was 1⁄6 of people with sixes, and 5⁄6 of people without sixes. Then the angel rearranged people, so the measure of people with sixes was increased to 1⁄2, and the measure of people without sixes was decreased to 1⁄2. Now at the end, your probability remains 1⁄6, so the other person’s probability is 5⁄6. -- The confusing part is that the other person at the beginning also had chance 1⁄6 to have 6. But if they had 6 and you didn’t, then there is a higher probability that the angel will assign them to you.
In other words, you should believe that their chance is 5⁄6 in the sense they if they didn’t roll 6, they would be more likely to be assigned by the angel to someone who did; but instead they were assigned to you (and you with probability 5⁄6 are a person who didn’t roll 6).
And the same holds for the other guy?
Or he must employ a different logic here?
Yes, it does.
It is important to remember that first we rolled dice, and only later the angel decided that we two will be a pair. If both of us would not roll 6, the angel would simply decide to not bring us together. -- The paradox of infinity is that the angel will always have enough other people to bring, if we both happen to not roll 6.
Let’s make it simple by removing the infinities.
There are exactly one million people, each of them rolls a die. An angel blindfolds them, and sorts them into two large groups: those who rolled a six, and those who didn’t. The blindfolded people cannot compare the sizes of their groups. Then the angel asks you: “What is the probability of you having rolled six?” (You can assume that the angel speaks simultaneously with everyone, so there is nothing special about asking you this question, instead of asking it someone else.)
A logical answer would be: 1⁄6.
Then the angel says: “Now I choose a random person from the other group; let’s call him Joe. What is the probability that Joe has rolled a six?”
A logical answer: 5⁄6.
The angel says: “However, Joe believes his chance of having rolled a six is 1⁄6, and he uses the same reasoning that you do. Is he wrong or what? Remember that I am having this dialogue simultaneously with each participant of the experiment.”
A logical answer: Probability is in the mind. In my mind, Joe has a probability 5⁄6 of having rolled a six. In Joe’s mind, he has a probability 1⁄6 of having rolled a six. These are two different questions.
Imagine that there are six people: me, Joe, Adam, Betty, Cecil, Daniel; and exactly one of us has rolled a 6. If it was me who rolled the six, you could have chosen any of {Joe, Adam, Betty, Cecil, Daniel} to be my twin. But if it was Joe who rolled the six, you didn’t have another choice for my twin, only Joe. So the fact that Joe is my twin, is for me an evidence that Joe has rolled the six. And symetrically, if you chose me as a twin for Joe, that is for him an evidence that I have rolled the six.
Because we have this finite scenario, one person was chosen five times as a “twin”, one person was chosen once, and four people we not chosen as “twins” at all. So we have only one of six people rolling 6, but we have five of six “twins” rolling 6. Therefore my probability of rolling 6 is 1⁄6 and my “twin”’s probability is 5⁄6.
...however in the infinite version, even if one subset is 5 times smaller than the other, the angel can choose a twin for everyone. But in some sense it means that if you rolled six, you are five times more likely to be selected as someone’s pair, as you would be if the angel just ignored the numbers on dice and paired people randomly.
While they are two different questions, you both have the same evidence and the same priors. They are symmetric questions, and should have identical answers.
So, I have 1⁄6 probability for the 6, he has 1⁄6 probability for the 6 and one of us has the 6 for sure?
Yes.
[ETA]: It’s important to remember that the probabilities are relative; you have odds of 1⁄6, you should posit odds of 5⁄6 for your twin. Likewise, your twin has odds of 1⁄6, and your twin should posit odds of 5⁄6 for you.
Say, that the 6 is worth US$1. What is the fair price to sell it, for those two? 20 cents for sure?
In that case the buyer would profit 60 cents.
Are you selling the outcome? I/e, “If I rolled a 6, you can have the $1”?
Because yes, both players should choose to sell the option. And likewise, both should choose to buy if their twin offers. Or, alternatively, both players should rationally agree to exchange outcomes.
And precisely because of how the angel assigned the twins, people would be winning on average in this game.
With infinities, this is possible. Just like in the infinite world, pyramid schemes are possible. You can generate free money simply by having each person with number N send $100 to a person with number N/2.