Make the game last at most an hour with exponentially decreasing round lengths.
Assuming you mean that the game is guaranteed to be completed within one hour, it is equivalent to a potentially infinitely long game with fixed round length, and we are back to my point.
When she wakes up, she knows the game has ended. Before she goes to sleep, she knows she will know this when she wakes up. So how can her probability estimate change between the two events?
When she goes to sleep, doesn’t she not know whether you will play the game? After all, the number of ‘players’ depends on how many rounds it will take before double sixes are rolled so the only guaranteed way for you to be picked is if you were first in line to be picked.
When she goes to sleep, she believes you have a 1⁄36 chance of dying.
When she wakes up, she believes you have a >90% chance of being dead.
But the only thing she learns on waking up is that the game has finished, and when she went to sleep she knew that she would learn this. This violates the maxim “if you know your destination, you are already there”.
Different levels of knowledge result in different probabilities… You have more information than she does. Your calculation computes the prior probability of you dying during one round. Her calculation computes the posterior probability of you dying during the whole game, given that you played and that the game has ended. The paradox only arises if one treats probability as an objective thing.
When you’ve just entered the room, what knowledge does one of you have that the other doesn’t; or why are you computing the probabilities of different events?
She has (will have) the knowledge that the game has ended, you don’t. If you could know that the game ends in this round, your probability of dying would be 100%.
She knows the game will end / will know the game has ended. She doesn’t know which round it will end / ended in. You also know the game will end, and not in which round; and you know what she will know upon waking up.
Looks like we are talking past each other, so the only way to continue is to show the calculation:
yours:
P(you die in this round| you play this round) = 1⁄36
your mom’s:
P(you die|game is over) = P(you play in the last ever round) = 9⁄10
You can express P2 by summing P1 over multiple rounds, weighted by the odds of the round being last and by the odds of you playing in it. But the important point that P1 and P2 are probabilities of different events.
Assuming you mean that the game is guaranteed to be completed within one hour, it is equivalent to a potentially infinitely long game with fixed round length, and we are back to my point.
When she wakes up, she knows the game has ended. Before she goes to sleep, she knows she will know this when she wakes up. So how can her probability estimate change between the two events?
When she goes to sleep, doesn’t she not know whether you will play the game? After all, the number of ‘players’ depends on how many rounds it will take before double sixes are rolled so the only guaranteed way for you to be picked is if you were first in line to be picked.
I’m supposing that she goes to sleep right after you’ve been chosen to play (i.e. you’ve been called into the room, the dice are about to be rolled).
If she knows that the game will have ended, she knows that the odds will be grim. Not sure what two events you are talking about.
I think you are claiming that
When she goes to sleep, she believes you have a 1⁄36 chance of dying.
When she wakes up, she believes you have a >90% chance of being dead.
But the only thing she learns on waking up is that the game has finished, and when she went to sleep she knew that she would learn this. This violates the maxim “if you know your destination, you are already there”.
Err… No, my point was quite the opposite.
Oh, I see. In that case, when you enter the room, why is her probability estimate different from yours? (Or if it’s not, why is yours >90%?)
Different levels of knowledge result in different probabilities… You have more information than she does. Your calculation computes the prior probability of you dying during one round. Her calculation computes the posterior probability of you dying during the whole game, given that you played and that the game has ended. The paradox only arises if one treats probability as an objective thing.
When you’ve just entered the room, what knowledge does one of you have that the other doesn’t; or why are you computing the probabilities of different events?
She has (will have) the knowledge that the game has ended, you don’t. If you could know that the game ends in this round, your probability of dying would be 100%.
She knows the game will end / will know the game has ended. She doesn’t know which round it will end / ended in. You also know the game will end, and not in which round; and you know what she will know upon waking up.
Looks like we are talking past each other, so the only way to continue is to show the calculation:
yours:
P(you die in this round| you play this round) = 1⁄36
your mom’s:
P(you die|game is over) = P(you play in the last ever round) = 9⁄10
You can express P2 by summing P1 over multiple rounds, weighted by the odds of the round being last and by the odds of you playing in it. But the important point that P1 and P2 are probabilities of different events.
And with that I am disengaging.