The Irksome Tuesday Boy Problem
Find a realistic scenario where this might happen.
03 November 2013
A few years ago, at a Gathering for Gardner puzzles and games convention in the USA, a man called Gary Foshee announced:
I have two children, (at least) one of whom is a boy born on a Tuesday - what is the probability that both children are boys?
I've put the words 'at least' in brackets because I don't think those words were actually used, but they should have been.
The Tuesday-boy puzzle went viral and caused a huge amount of debate when it was declared that the answer is 13/27, or just under half. Why isn't the chance of the other child being a boy simply 50-50, and what on earth has Tuesday got to do with it?
This puzzle has always irked me. Why? Because it doesn't have a 'right' answer. It all depends on what led to the person making the statement about the boys in the first place.
In what follows I will ignore the Tuesday reference since it makes the calculations much more complicated. If you want to know why Tuesday can change the answer, look for explanations online, for example by Alex Bellos.
Let's just concentrate on the basic problem. Somebody says: "I have two children, (at least) one of them is a boy. What is the chance that both are boys?". The standard solution is that the chance that both children are boys is 1 in 3. The reasoning goes like this. Since you know I have at least one boy, there are three possible pairs of children I might have (older then younger):
Each pairing is equally likely, and in only one of these three cases is the 'other' child also a boy.
But 1/3 is not the only valid answer. To see why, let's simulate the occurrence of boys and girls by tossing a coin.
I have just flipped two coins. At least one is a Head. What is the chance that both are Heads?
You might argue that since we have just replaced Boy with Head, it is the same puzzle and so the chance of the 'other' coin being a Head is 1 in 3. The good news is that we can test it for real by flipping the coins 100 times and seeing how often the two coins are the same. If the odds really are 1 in 3, then we should find that the two coins are the same roughly 33 times.
What you will discover when you play the game with me, however, is that the coins are the same about 50 times out of 100, not 33. Why? When I flip two coins, my rule is that I always look at one of them at random. If it is a Head I then say "I have two coins, at least one is a Head". If I see a tail first I say "At least one is a Tail".
When somebody says to you "I have two children, at least one of them is a boy", who is to say that they weren't using exactly the same rule as me, picking one child at random and revealing its gender. In fact, I reckon that this is the more likely way in which such information would be revealed in a real life situation. In which case, the answer 1/2 is more right than 1/3.
Incidentally, there is also a scenario in which there is a zero probability of the two children both being boys. If the word 'one' is emphasised, then in the statement "I have two children, at least one of them is a boy", the words "at least" really mean "thank goodness". "At least ONE of the children is a boy" ("because if both were girls we would have to pay for TWO weddings").
And in the Tuesday boy problem, if the word Tuesday is emphasised, then the chance of both children being boys is arguably 100%. "I have two children, one of them is a boy born on a TUESDAY...." (in contrast to the other boy who was born on a WEDNESDAY).
Most of my concern with the way the puzzle is usually posed is that it is the parent of the children who 'spontaneously' reveals the information about their child. The ambiguity could be removed if it is you, the puzzle solver, who questions the parent. You bump into somebody in a shopping mall and ask them:
"Do you have two children?"
"Yes", they reply.
"Is at least one of them a boy?"
Set up this way, I agree that there is a 1/3 chance that the other child is a boy.
I will end this blog with a challenge. Let's call these situations: "There are two Xs, at least one is a Y." Can anybody find a realistic situation that might genuinely arise - or even better, that has actually happened to you - where the probability that both Xs are Ys really IS 1/3? Email me your suggestions at email[AT]robeastaway.com.
[July 2017 update] Well in the three years since this blog was published, I still haven't had anybody give me a realistic scenario in which the 1/3 outcome has cropped up.* In an episode of 'The Infinite Monkey Cage' on BBC Radio 4, Richard Wiseman posed the puzzle in minimalist form: "he has two children, one of them is a boy", leaving out the "at least", just as Gary Foshee did. Unsurprisingly many who heard it were cross that the answer given in the next episode was 1/3.
* Rupert Yardley emailed to make the case that in insurance, one might need to use 'Tuesday boy' logic to ensure the premiums are set at the correct rate. He offered an interesting example involving an all-boys school being given a special offer for siblings. But it was still an invented scenario, not one that's actually happened.