At the end of Monday's IRC, flagamuffin and wasoxygen were briefly discussing http://crackthesenuts.blogspot.com/2013/12/born-on-tuesday.html I really love these problems and wanted to discuss, but I missed you guys.
The question was: "Suppose I tell you that I have two children and one of them is a boy, what is the probability that I have two boys?" He claims the answer is 1/3, but I think it depends on why he is telling us that he has a boy.
If he was thinking of one of his children, and told us the gender of that child, we have no information on the other child, and the probability remains at 1/2. If instead we had asked "Do you have at least one boy?", and he replied "Yes", the probability is 1/3 that he has two.
The lecture in the link goes over some similar problems, and provides a framework for dealing with them. He uses it to explore some pretty interesting questions, like the probability of humanity lasting another ten thousand years.
I enjoyed The Accidental Universe, much of which dealt with the anthropic principle -- though not in nearly as much depth. As an aside, can any of you tell me why I've heard of Scott Aaronson, other than perhaps having read his website before? I can't remember. His Wikipedia page didn't help. Anyway, badge for pure fascination. What a great read this is. I think if I badge it the probability that more hubski users will click rises. -- I enjoyed going over the doomsday argument in my mind. However, I think it is significant that the probability of snake eyes at any given iteration of dice rolls is quite small -- less than 3 percent. As one of the commenters points out (and as Scott acknowledges), assuming infinite or finite people/rolls/population/etc changes things. Hmm. More thinking to come. I am inclined toward SIA -- existence as evidence.
Wow, thanks for the badge and the book recommendation. I've been (passively) looking for a good arrow of time exploration, and it seems like The Accidental Universe might have one. I'm not sure where you would have heard of Scott Aaronson, I had previously enjoyed his "Who Can Name the Bigger Number", but didn't make the connection until reading his Wikipedia page. Perhaps you're thinking of Scott Adams? I think that's why his name seems so familiar to me. -- I think wasoxygen had a point about the dice room only seeming paradoxical due to the ill-defined infinities involved. I can't come up with a finite game with the same principles that seems counter-intuitive. A commenter in the article wasoxygen linked mentioned the problem is similar to the roulette strategy where you keep doubling your bet until you win. Even if these do all boil down to imprecise language, I've really enjoyed playing with the ideas.
I am not thinking of Scott Adams. I used to have a high opinion of Scott Adams and now I don't. I like the roulette take, and the concept of an immortal making a bet with each person who enters the doomsday room. There are many fascinating ways to think about this.
It's strange to me that God's Debris would have lowered your opinion of Scott. Much of the criticism in that thread seemed based around flaws in Avatar's assertions/logic, which I thought was sort of the point.
I'm only a third of the way through, but he seems to just be giving a brief introduction to a philosophical concept he finds interesting, and then providing a simple "answer" to seed thought. My main complaint so far is that I'm already familiar with most of the concepts he brings up, and I'm not finding many of the seed answers novel or compelling. However, I would have loved this 5-10 years ago. I've enjoyed thinking about these concepts as I've come across them. It doesn't really seem like Scott's fault that I'm no longer in the target audience.The central character states a number of scientific “facts.” Some of his weirdest statements are consistent with what scientists generally believe. Some of what he says is creative baloney designed to sound true. See if you can tell the difference.
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."