On the “Son born on a Tuesday” probability puzzle
If someone tells you “I have two children. One of them is a boy born on a Tuesday”, what is the probability that he has two sons?
The only thing we need in order to solve this problem is an interpretation. Here is the most natural one.
There is a crowd of people who have exactly 2 children. We pick one person at random and ask “do you have a son born on a Tuesday?”, and then ask “do you have 2 boys?”. If the answer to the 1st question is “yes”, what is the probability that the answer to the 2nd question is “yes”? (I omit the trivial assumptions about all outcomes being equally probable).
Once we have this interpretation, we can give this problem to any kid who knows how to multiply 14 by 14.
Here is a scan of the solution that my son (born on a Monday) came up with. Please pardon his chicken scratch:
The filled squares are families that have a tuesday boy. The top left quadrant are boy-boy families. The answer is the ratio of the number of filled squares in that quadrant to the number of all filled squares. That is 13/27.
Yes, I explained to him about conditional probabilities and gave some examples, but only to show why such problems are so unintuitive.
"Tuesday son" is an anagram of “so unsteady” and “yes astound”.
I have to confess that the purpose of this post is to test the hypothesis that blog posts that talk about this problem attract a lot of traffic. So in case it turns out to be true: dear reader, you may also find the following topics interesting:
https://bbzippo.wordpress.com/2010/06/02/an-unfair-game/ – guessing hidden numbers using a random generator
https://bbzippo.wordpress.com/2010/05/19/optional-stopping/ – a theorem about timing the market
https://bbzippo.wordpress.com/2009/11/05/expected-number-of-coin-tosses/ – What is the average number of coin tosses needed to throw a head?
https://bbzippo.wordpress.com/2009/11/25/expected-number-of-tosses-general/ – What is the average number of coin tosses needed to throw n heads in a row?
https://bbzippo.wordpress.com/2009/11/05/probability-of-bias/ – making a conclusion about the coin being biased based on a single throw.
And don’t forget to check out Xworder 🙂