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.
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.