# Puzzling Adventures: A Kingdom for My Child

Looking for the complete family

Image: Cloe Liane Shasha

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.

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1. 1. helpdeskgeek 02:08 PM 8/1/08

The answer to #5 can't be right. You are double- and sometimes triple-counting those families with more than one daughter.

2. 2. helpdeskgeek 02:09 PM 8/1/08

The answer to #5 can't be right. You are double- and triple- counting the families with more than one daughter.

3. 3. btvdan in reply to helpdeskgeek 02:51 PM 8/4/08

Hi "helpdeskgeek": I posted this note to the discussion on the answer page. I've always known myself to be extremely challenged with probability... one the things that probably helps with this is to ask, "how would we test this in an experiment or collect the statistics in real life?" Anyhow, this is my question --> I never understand how to count these problems. Maybe someone can explain it better for me. The question asked, "...then what was the likelihood that her family also had a boy?" We are interested in counting the total number of families that the girl could belong to and then selecting the number of families from the selection space that have the interesting characteristic (the family has a boy). This leads me to the 6/7 result. It seems to me that 7/10 answers a different question (the likelihood that the girl has a brother). It seems the fact that the girl comes to school only serves to eliminate 3 boy families from consideration. This also asks the question... can the family be more likely to have a boy than a girl have a brother (when considering both within this same framework of rules).

Is there a better way to explain this? Considering the 10,000 girls type thinking doesn't help me because it sounds like the other question to me. Thanks!

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