It's quite an interesting lecture. I get a little queasy trying to comprehend and think through the anthropic principle. The comments on the article were good, and I especially liked one from (Hugo-winner) Greg Egan. In the comments (with spoilers) to "Born on a Tuesday" (your URL includes a stray period) I observed that This would be a very peculiar understanding of what "having two children" means. Such lateral thinking can be very helpful in solving contrived puzzles, but I think we should keep our language as clear and grounded as we can when trying to figure out the real world.the answer to the first question can be 1/2 instead of 1/3 with a particular understanding of the statement. If the person "has" two children because they picked them at random from a very large supply of children, half boys and girls, and they have only looked at one child to determine that he is a boy, there is a 1/2 chance that the second child is also a boy.
I hadn't seen the comments on the article, thanks for pointing them out. That's a pretty impressive readership. I had seen Steve's comment, but I disagree with him that 1/2 is the solution that requires a contrived understanding of the puzzle. The 1/3 solution is only valid if we know he will preferentially reveal a male child over a female. Imagine Steve wants to pose this puzzle. He has two children. There is a 25% chance he has two boys, so he would be forced to say "...one of them is a boy...". There is a 25% chance he has two girls, in which case he must say "...one of them is a girl...". There is a 50% chance that he has both a boy and a girl, in which case he can pick either variation of the puzzle. We heard him use the male version. If we know he would always choose the male version when given a choice, the answer would be 1/3. If we know he would prefer the female version, he must have two boys. If he has no known bias towards one or the other, we gain no additional knowledge, and the probability remains 1/2.
I agree, the speaker's intentions affect the result. With this as a given, we are in trouble already. We should probably assume that Steve will choose language that he thinks most likely to lead us to the wrong answer. Consider a casual dialog between office workers: The question "How long is a day?" has different answers when it is posed by a child, an astronomer, and a puzzler. The language used in discussing the anthropic principle seems very intentional and motivated, and therefore possibly misleading. Commenter Russ Gorman on another discussion makes what I think is a valid point: the possibilty of never throwing snake-eyes in the Dice Room. This may be a remote chance when there are infinite trials, but not as remote as having an infinite supply of humans to kidnap. I found this discussion when searching for clarification on whether the Dice Room (or Shooting Room) uses selection with replacement, which would clearly affect your odds of survival in the long run.Imagine Steve wants to pose this puzzle.
"So, do you have any kids?" "Yeah, two." "Oh? Any boys?"
When that last sentence appears outside a puzzle blog or MIT lecture the most reasonable conclusion is that the speaker has one and only one boy. "One of them is a boy."