In the ancient society of Machudo, families wanted no more than three kids. Their eldest son had a chance of becoming king, so they would stop having children after they had their first boy. A family that had three children or had a boy was said to be "complete."

Assume boys and girls had an equal probability of being born. (In reality, boys are slightly more likely, but it's undignified for puzzle masters to deal with slight exceptions to basic rules.)

Warm-up:
What fraction of complete families would have a boy?

Now for the actual puzzle. Probability problems can be difficult to model mathematically. That's a nice way to say that many people get their models wrong. The simplest way to get the model right is to enumerate all the equally probable outcomes and count the proportion that are in each class of interest. You may need a computer to do this in general, but not for the puzzles below. (Manda Wilson and I have just completed a book called Statistics is Easy! extending these ideas to situations where you might need a computer.)

Problems

1. What was the average number of children per complete family?

Whereas families still wanted no more than three children, Chiwachi, the king of Machudo, one day decided that queens should be allowed as well as kings. So, he decreed that either the first born son could be a king or the first born daughter could be queen and this would be based on a magic ritual that either sex could win. This became known as the Chiwachi rule.

People understood that the ideal family had become one with both a boy and a girl. Thus, a family was now complete if it had three children or it had at least one boy and at least one girl. We call such a family Chiwachi-complete. Every family eventually becomes Chiwachi-complete.

2. What fraction of Chiwachi-complete families would have at least one boy and at least one girl?

3. What was the average number of children per Chiwachi-complete family?

4. If you knew that a Chiwachi-complete family had three children and you heard that the youngest child is a girl, then what was the likelihood that the family had at least one boy?

School was mandatory for all children in Machudo.

5. If you saw a girl enter school and all you knew was that she came from a Chiwachi-complete family (but you didn't know her birth order), then what was the likelihood that her family also had a boy?

Dennis Shasha is at the Courant Institute of Mathematical Sciences, New York University. His most recent puzzle book, Puzzles for Programmers and Pros, was published in May 2007 by John Wiley and Sons/Wrox.