(Note: I got the numbers slightly wrong—the 1001s should have been 1000s etc.)
Yes: If the coin was heads then the probability of event “clone #707 is in a green room” is 1/1000. And since, in this case, the clone in the green room is sure to be an anthropic reasoner, the probability of “clone #707 is an anthropic reasoner in a green room” is still 1/1000.
On the other hand, if the coin was tails then the probability of “clone #707 is in a green room” is 999/1000. However, clone #707 also knows that “clone #707 is an AR”, and P(#707 is AR | coin was tails and #707 is in a green room) is only 1⁄999.
Therefore, P(#707 is an AR in a green room | coin was tails) is (999/1000) * (1/999) = 1/1000.
If the coin was heads then the probability of event “clone #707 is in a green room” is 1/1000. And since, in this case, the clone in the green room is sure to be an anthropic reasoner, the probability of “clone #707 is an anthropic reasoner in a green room” is still 1/1000.
But you know that you are AR in the exact same way that you know that you are in a green room. If you’re taking P(BeingInGreenRoom|CoinIsHead)=1/1000, then you must equally take P(AR)=P(AR|CoinIsHead)=P(AR|BeingInGreenRoom)=1/1000.
and P(#707 is AR | coin was tails and #707 is in a green room) is only 1⁄999.
Why shouldn’t it be 1/1000? The lucky clone who gets to retain AR is picked at random among the entire thousand, not just the ones in the more common type of room.
(Note: I got the numbers slightly wrong—the 1001s should have been 1000s etc.)
Yes: If the coin was heads then the probability of event “clone #707 is in a green room” is 1/1000. And since, in this case, the clone in the green room is sure to be an anthropic reasoner, the probability of “clone #707 is an anthropic reasoner in a green room” is still 1/1000.
On the other hand, if the coin was tails then the probability of “clone #707 is in a green room” is 999/1000. However, clone #707 also knows that “clone #707 is an AR”, and P(#707 is AR | coin was tails and #707 is in a green room) is only 1⁄999.
Therefore, P(#707 is an AR in a green room | coin was tails) is (999/1000) * (1/999) = 1/1000.
But you know that you are AR in the exact same way that you know that you are in a green room. If you’re taking P(BeingInGreenRoom|CoinIsHead)=1/1000, then you must equally take P(AR)=P(AR|CoinIsHead)=P(AR|BeingInGreenRoom)=1/1000.
Why shouldn’t it be 1/1000? The lucky clone who gets to retain AR is picked at random among the entire thousand, not just the ones in the more common type of room.
Doh! Looks like I was reasoning about something I made up myself rather than Jordan’s comment.