A few days ago I posted a probability brain teaser which was getting me in a spin.
A family has 2 children. When you knock on the door one of the children, a boy, opens the door. What is the probability that the other child is a boy?
There were many interesting responses and as I expected, I had a mixture of two answers: a third and a half.
That which gives an answer of a third
When I first told this problem over dinner, I was told the answer was a third. The explanation is as follows:
You know that a boy opened the door. That means there is at least one boy. So out of the four possible permutations: boy-boy, boy-girl, girl-boy, girl-girl only the first three are possible. You could essentially rephrase the question as “There are two children in the family and it is not the case that both are girls”.
In the three of the permutations which are now possible (boy-boy, boy-girl, girl-boy), only one gives two boys. So the likelyhood of there being two boys is 1/3rd. Hence, the probability of the other child being a boy is a third.
That which gives an answer of a half
My immediate response was that the gender of the two children are independent and that the gender of one has no bearing on the other.
I also pursued a different technique which also lead to the answer of a half. There are two children in the family: giving the possible combinations of boy-boy, boy-girl, girl-boy and girl-girl. The probability of each is 25% or 1/4.
There are three possible ways that could lead to a boy opening the door. If there are two boys in the family then a boy will most definitely open the door. If there is one boy and one girl in the family, the chance that the boy will open the door in each case is only a half.
I’ve illustrated this information in a probability tree diagram where the probabilities shown are cumulative.
The total probability of a boy-boy combination with a boy opening the door is 25%. The total probability of there being one boy and one girl, and a boy opening the door is also 25%.
The probability of a 1 boy, 1 girl combination is hence the same as the probability of 2 boys.
After thinking both of these arguments through, I can’t actually see the logical fallacy which exists in either. They can’t both be right however. It would be great to hear your thoughts once again!