In the class I’m teaching we’ve entered probability. This is a fun subject. It’s one of the bits of mathematics which people encounter most often, about as much as the elements of geometry enter ordinary life. It seems like everyone has some instinctive understanding of probability, at least given how people will hear a probability puzzle and give a solution with confidence. You don’t get that with pure algebra problems. Ask someone “the neighbor’s two children were born three years apart and twice the sum of their ages is 42; how old are they?” and you get an assurance of how mathematics was always their weakest subject and they never could do it. Ask someone “one of the neighbor’s children just walked in, and was a girl; what is the probability the other child is also a girl?” and you’ll get an answer.

But it’s getting a correct answer that is really interesting, and unfortunately, while everyone has some instinctive understanding and will give an answer as above, there’s little guarantee it’ll be the right one. Sometimes, and I say this looking over the exam papers, it seems our instinctive understanding of probability is designed to be the wrong one. I’m happy that people aren’t afraid of doing probability questions, not the way they are afraid of algebra or geometry or calculus or the more exotic realms, though, and feel like it’s my role to find the most straightforward ways to understanding which start from that willingness to try.

Some of the rotten track record people have in probability puzzles probably derives from how so many probability puzzles start as recreational puzzles, that is, things which are *meant* to look easy and turn out to be subtly complicated. I suspect the daughters-question comes from recreational puzzles, since there’s the follow-up question that “the elder child enters, and is a girl; what is the probability the younger is a girl?” There’s some soundness in presenting the two as a learning path, since they present what looks like the same question twice, and get different answers, and learning why there are different answers teaches something about how to do probability questions. But it still feels to me like the goal is that pleasant confusion a trick offers.