Jason Kottke posts this puzzle from Gary Foshee that reportedly impressed people at a puzzle-designers’ convention:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The first thing you think is “What has Tuesday got to do with it?” Well, it has everything to do with it.
I thought I should really figure this one out myself before reading any further, and I decided this was a good time to apply my general principle that it’s always best to solve such problems from scratch rather than trying to guess at the answer.
So I laid out all the 4 x 49 possibilities. The 4 is bb, bg, gb, gg, and the 49 are all possible pairs of days of the week. Then I ruled out all the possibilities that were inconsistent with the data: this leaves the following:
bb with all pairs of days that include a Tuesday. That’s 13 possibilities (Mon/Tues, Tues/Tues, Wed/Tues, …, Tues/Mon, …, Sun/Tues, remembering not to count Tues/Tues twice).
bg with all the Tues/x pairs: that’s 7 more possibilities.
gb with all the x/Tues pairs: that’s 7 more.
Next, the assumptions. I decided I’d keep it simple:
– Ignore multiple births.
– Pretend that boys and girls are equally likely. (Regular readers know that Pr (girl) = .485.)
– Pretend that births are equally likely on every day. (Actually, I seem to recall reading that they’re more common on Fridays and less so on weekends.)
Pr (bb | available information) is then 13/27.
But then I thought, Hey, he said, “One is a boy born on a Tuesday.” He didn’t say At least one. So I’ll toss out bb Tues/Tues, which leaves us with 26 possibilities and a conditional probability for bb of 12/26.
Having solved the problem to my satisfaction (but with a bit of a worry that I was missing something important), I followed the link to a news article by Alex Bellos, who gives the answer as 13/27. So I guess when Foshee said “One,” he meant, “At least one.”
Perhaps the best comment on all of this, however, is from Todd Stark, who writes:
Doesn’t this illustrate limits to the value of probability? It seems like more than a curiosity to me. If specifying a logically irrelevant detail changes the probability calculation, doesn’t that tell us that probability thinking is a relatively useless tool in situations like this? It is implicit that everyone is born on a particuar day, if specifying something we already knew changes the calculation, isn’t the calculation unreliable for decision making, for this class of situations?
Good question. I certainly believe that probability theory is the right mathematical tool for solving probability problems–as a statistician, I guess it’s no surprise that I feel this way–but, given how difficult it is to solve such problems in one’s head, it’s hard to see this as a useful model for regular decision making.
The interesting question, I think, is how often do these sorts of tricky conditional probability problems arise in real life. I don’t know the answer. (That is, I’m not trying to raise a rhetorical question and claim that these problems don’t arise in real life. What I’m saying is that I don’t know and would be interested in seeing how to think systematically about the question.)
P.S. Bellos’s article was fine, but I wish he’d remarked that these conditional probability examples are textbook problems in introductory probability courses.
P.P.S. I agree with the many commenters who point out that, really, the information to condition on is not “Foshee has two children. One is a boy born on a Tuesday,” but, rather, “Foshee says, ‘I have two children. One is a boy born on a Tuesday.'” So really you need a model for what Foshee might be saying.
That’s one reason I’m not a big fan of this sort of trick probability question: some of the most important parts of the problem are hidden, and the answer is typically explained in a way that avoids making clear the assumptions that are needed to get there.