Anthropic evidence depends on the reference class you put yourself into, which in this case has a very definite meaning: “What kind of other person would be faced with the same problem you are?”
If you play the game for up to 1000 rounds, then with probability 1⁄2 − 1/2^1000 you got a T at some point and survived anyway (we call this a Type 1 outcome). Otherwise, you are probably dead, but with probability 1/2^999, you saw a run of 1000 heads (a Type 2 outcome).
It’s true that Pr[gun is loaded | you survived] is close to 0. In fact, if you didn’t see the coin flips then this is the probability you should use: given that you survived 1000 coin flips, you are better off continuing the game, because the gun most likely isn’t loaded.
But once you get shown the coin flips, then the decisions you can make are different. In a Type 1 outcome, you know the gun isn’t loaded, because it’s been triggered and didn’t kill you (Edit: so this means you’re free to leave anyway). So when we decide “what should we do if the coin came up “heads” 1000 times?” we should only be looking at the Type 2 outcomes, because that’s the only situation in which we need to make that decision.
And the Type 2 outcome happens with equal probability whether or not the gun was loaded. Therefore you should expect the gun to be loaded with 50% probability, and keep playing.
Anthropic evidence depends on the reference class you put yourself into, which in this case has a very definite meaning: “What kind of other person would be faced with the same problem you are?”
If you play the game for up to 1000 rounds, then with probability 1⁄2 − 1/2^1000 you got a T at some point and survived anyway (we call this a Type 1 outcome). Otherwise, you are probably dead, but with probability 1/2^999, you saw a run of 1000 heads (a Type 2 outcome).
It’s true that Pr[gun is loaded | you survived] is close to 0. In fact, if you didn’t see the coin flips then this is the probability you should use: given that you survived 1000 coin flips, you are better off continuing the game, because the gun most likely isn’t loaded.
But once you get shown the coin flips, then the decisions you can make are different. In a Type 1 outcome, you know the gun isn’t loaded, because it’s been triggered and didn’t kill you (Edit: so this means you’re free to leave anyway). So when we decide “what should we do if the coin came up “heads” 1000 times?” we should only be looking at the Type 2 outcomes, because that’s the only situation in which we need to make that decision.
And the Type 2 outcome happens with equal probability whether or not the gun was loaded. Therefore you should expect the gun to be loaded with 50% probability, and keep playing.