After several weeks of heavier items, here is a puzzle just for the fun of it. I was challenged by it, but instead of deriving the answer from scratch, I recognized it as being related to a probability puzzle we have featured before in several modes. Here is the puzzle, followed by the related one that helped me to quickly solve it, and finally the “from scratch” solution. No peeking.

Assume you have a friend you have not seen in years, but you know that in the intervening time he has grown a family with three children. You do not know their names or genders. Your friend invites you to visit. When you arrive, you are met at the door by a boy who says you must be his father’s friend. He is the only child at home at that time. He says that his parents are expecting you and that his siblings will arrive later. He is polite and takes you through the house to the back patio where his parents are enjoying the summer afternoon outside. As you pass through the kitchen, you notice a picture on the refrigerator. It is a girl in school clothes. You realize from her family resemblance that she must be the boy’s sister — one of your friend’s three children.

As you greet your friend and meet his wife for the first time, you wonder whether the third child is a boy or girl. Since you saw the boy and a picture of a girl, he has either two boys and one girl or two girls and one boy. What is the probability of either combination? In the absence of extraneous factors such as a genetic disposition favoring boys or girls, it seems obvious that the probabilities are equal. Is that right?

This is a puzzle that seems more complicated than it is. When you strip away some of the narrative, you might recognize it as a variation of the puzzle of two children (or two coins, whatever) where the probabilities of some feature (boy or girl: head or tails) is independent for each event.

In the more common version (and more easily understood), a man says he has two children. At least one is a boy. What is the probability the other is a boy? Naively one might answer that the probability of either gender is 0.5, but by the conditions of the puzzle, the children could be BG, BG, or BB with the GG combination ruled out. Or the three possible combinations, in two of them the second child is a girl. So the probability of the second child being a girl is twice that of being a boy.

This solution seems counter-intuitive. However, if we change the puzzle slightly and say the oldest child is a boy, then the probability changes. Why?

With that simpler puzzle in mind, we see that the same analysis holds for the case of three children. The possible combinations are BBG, BGB, or BGG with the possibility of GGG being excluded by the condition of having seen at least on boy child. Therefore the probability of the third child being a girl is again twice that of being a boy.

How would this analysis change if your friend greeted you and said, “I see you have met my youngest child” as the boy escorted you to through the kitchen?

Suppose he then said that his oldest child does not play sports, but his daughter likes soccer. What are the probabilities then?

The clues in all these variations depend on whether events are independent or linked in some way. Sometimes a subtle change in wording can significantly change the probability of an event. Clever gamblers can make a lot of money on this.

*In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.*

[tags]statistics, probability, chance, puzzle[/tags]