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?