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.
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.