I think you’re wrong. Suppose 1,000,000 people play this game. Each of them flips the coin 1000 times. We would expect about 500,000 to survive, and all of them would have flipped heads initially. Therefore, P(I flipped heads initially | I haven’t died yet after flipping 1000 coins) ~= 1.
This is actually quite similar to the Sleeping Beauty problem. You have a higher chance of surviving (analogous to waking up more times) if the original coin was heads. So, just as the fact that you woke up is evidence that you were scheduled to wake up more times in the Sleeping Beauty problem, the fact that you survive is evidence that you were “scheduled to survive” more in this problem.
On the other hand, each “heads” you observe doesn’t distinguish the hypothetical where the original coin was “heads” from one where it was “tails”.
This is the same incorrect logic that leads people to say that you “don’t learn anything” between falling asleep and waking up in the Sleeping Beauty problem.
I believe the only coherent definition of Bayesian probability in anthropic problems is that P(H | O) = the proportion of observers who have observed O, in a very large universe (where the experiment will be repeated many times), for whom H is true. This definition naturally leads to both 2⁄3 probability in the Sleeping Beauty problem and “anthropic evidence” in this problem. It is also implied by the many-worlds interpretation in the case of quantum coins, since then all those observers really do exist.
It’s often pointless to argue about probabilities, and sometimes no assignment of probability makes sense, so I was careful to phrase the thought experiment as a decision problem. Which decision (strategy) is the right one?
Actually you’re right, I misread the problem at first. I thought that you had observed yourself not dying 1000 times (rather than observing “heads” 1000 times), in which case you should keep playing.
Applying my style of analyzing anthropic problems to this one: Suppose we have 1,000,000 * 2^1000 players. Half flip heads initially, half flip tails. About 1,000,000 will get heads 1,000 times. Of them, 500,000 will have flipped heads initially. So, your conclusion is correct.
I think you’re wrong. Suppose 1,000,000 people play this game. Each of them flips the coin 1000 times. We would expect about 500,000 to survive, and all of them would have flipped heads initially. Therefore, P(I flipped heads initially | I haven’t died yet after flipping 1000 coins) ~= 1.
This is actually quite similar to the Sleeping Beauty problem. You have a higher chance of surviving (analogous to waking up more times) if the original coin was heads. So, just as the fact that you woke up is evidence that you were scheduled to wake up more times in the Sleeping Beauty problem, the fact that you survive is evidence that you were “scheduled to survive” more in this problem.
This is the same incorrect logic that leads people to say that you “don’t learn anything” between falling asleep and waking up in the Sleeping Beauty problem.
I believe the only coherent definition of Bayesian probability in anthropic problems is that P(H | O) = the proportion of observers who have observed O, in a very large universe (where the experiment will be repeated many times), for whom H is true. This definition naturally leads to both 2⁄3 probability in the Sleeping Beauty problem and “anthropic evidence” in this problem. It is also implied by the many-worlds interpretation in the case of quantum coins, since then all those observers really do exist.
It’s often pointless to argue about probabilities, and sometimes no assignment of probability makes sense, so I was careful to phrase the thought experiment as a decision problem. Which decision (strategy) is the right one?
Actually you’re right, I misread the problem at first. I thought that you had observed yourself not dying 1000 times (rather than observing “heads” 1000 times), in which case you should keep playing.
Applying my style of analyzing anthropic problems to this one: Suppose we have 1,000,000 * 2^1000 players. Half flip heads initially, half flip tails. About 1,000,000 will get heads 1,000 times. Of them, 500,000 will have flipped heads initially. So, your conclusion is correct.