The Left-Handed Sister Problem
Suppose you meet someone who looks like the brother of your friend Mary. You ask if he has a sister named Mary, and he says “Yes I do, but I don’t think I know you.”
You remember that Mary has a sister who is left-handed, but you don’t remember her name. So you ask your new friend if he has another sister who is left-handed.
If he does, how much evidence does that provide that he is the brother of your friend, rather than a random person who coincidentally has a sister named Mary and another sister who is left-handed? In other words, what is the Bayes factor of the left-handed sister?
- Out of 100 families with children, 20 have one child, 30 have two children, 40 have three children, and 10 have four children.
- All children are either boys or girls with equal probability, one girl in 10 is left-handed, and one girl in 100 is named Mary.
- Name, sex, and handedness are independent, so every child has the same probability of being a girl, left-handed, or named Mary.
- If the person you met had more than one sister named Mary, he would have said so, but he could have more than one sister who is left handed.
I’ll post a solution only when someone replies to this tweet with a correct answer!
If you like this sort of thing, you might like the new second edition of Think Bayes.