Here’s a modified Sleeping Beauty problem. Instead of having Sleeping Beauty awakened 2 times if the coin is tails and 1 time if the coin is heads, recruit two people, Sleeping Beauty and Snow White.
If the coin comes up heads, wake one of them randomly and ask the question, and just let the other one go.. If the coin comes up tails, wake both of them and then them both the question.
This version of the problem eliminates many of the troublesome factors of the original version, yet it it’s hard to justify why it would have a different answer than the original version. And the answer to this version is obviously that tails has a 2⁄3 probability.
Now, if you still think this has a different answer from the original problem, here’s yet another variation. You have a crowd of people, all of whom go to sleep. If the coin comes up heads you wake one of them and ask the question; if the coin comes up tails, you wake two of them and ask the question. Does the answer change if waking two of them happens with replacement (so you can pick the same person twice in which case you cause the same amnesia that’s in the original problem) or without replacement? And if the answer doesn’t change between with replacement and without replacement, then you should be able to shrink the size of the crowd down to 1 (thus reducing it to the original problem) while keeping the answer the same.
And if the answer doesn’t change between with replacement and without replacement, then you should be able to shrink the size of the crowd down to 1 (thus reducing it to the original problem) while keeping the answer the same.
Not so, if there is a crowd your being woken is stronger evidence of two people being woken than of one person being woken. In a crowd of 10, you have 1⁄10 chance of being woken if one random person is woken, and 19⁄100 chance of being woken at least once if two random people (with replacement) are woken. In a crowd of size 1, you have 100% chance to be woken at least once either way. Same odds == observation provides no evidence.
Don’t compute the odds that two people have been woken, compute the odds that this is a two-wakings experiment. That’s also higher than 50% and that (unlike “the odds that two people have been woken”) stays higher when you shrink the crowd size.
Here’s a modified Sleeping Beauty problem. Instead of having Sleeping Beauty awakened 2 times if the coin is tails and 1 time if the coin is heads, recruit two people, Sleeping Beauty and Snow White.
If the coin comes up heads, wake one of them randomly and ask the question, and just let the other one go.. If the coin comes up tails, wake both of them and then them both the question.
This version of the problem eliminates many of the troublesome factors of the original version, yet it it’s hard to justify why it would have a different answer than the original version. And the answer to this version is obviously that tails has a 2⁄3 probability.
Now, if you still think this has a different answer from the original problem, here’s yet another variation. You have a crowd of people, all of whom go to sleep. If the coin comes up heads you wake one of them and ask the question; if the coin comes up tails, you wake two of them and ask the question. Does the answer change if waking two of them happens with replacement (so you can pick the same person twice in which case you cause the same amnesia that’s in the original problem) or without replacement? And if the answer doesn’t change between with replacement and without replacement, then you should be able to shrink the size of the crowd down to 1 (thus reducing it to the original problem) while keeping the answer the same.
Not so, if there is a crowd your being woken is stronger evidence of two people being woken than of one person being woken. In a crowd of 10, you have 1⁄10 chance of being woken if one random person is woken, and 19⁄100 chance of being woken at least once if two random people (with replacement) are woken. In a crowd of size 1, you have 100% chance to be woken at least once either way. Same odds == observation provides no evidence.
Don’t compute the odds that two people have been woken, compute the odds that this is a two-wakings experiment. That’s also higher than 50% and that (unlike “the odds that two people have been woken”) stays higher when you shrink the crowd size.