Can you solve it? The two child problem

Here are four questions. They sound very similar. But be careful. They are not.

1. Mrs Smith has two children. The eldest one is a boy. What’s the chance that both are boys?

2. Mrs Jones has two children. At least one is a boy. What’s the chance that both are boys?

3. Mrs Robinson has two children. At least one is a boy born on a Monday. What’s the chance that both are boys?

4. Mrs Taylor has two children. At least one is a boy called Oscar. What’s the chance that both are boys?

(Assume Mrs Smith, Jones, Robinson and Taylor are each chosen randomly from the population of families with exactly two children. The phrase ‘at least one is a boy’ is understood in the literal sense, i.e. in this case either one child is a boy, or both children are boys.)

You might think that the boy’s day of birth, or indeed his name, are irrelevant. If you do, you are wrong. The answers are all different. There will be arguments. In preparation, I’m conducting an experiment. If you are the parent of exactly two children, please fill in the questionnaire below.

Read the four numbered statements. If any of them apply to you, then click the correct option below. If the statement does not apply to you, leave both options blank. (If you need to check the day of the week for a birthdate, click here.)

The experiment is – hopefully – going to reveal, and thereby confirm, the counter-intuitive solutions. For each question, the number of people clicking the second option divided by the total number of people clicking both options, should approximate the correct answer.

In order to get the largest possible data set, I’d be very grateful if you could get word out to any friends with exactly two children. (If you know any friends with two kids one of whom is called Oscar, submit on their behalf.) Thank you!

I’ll be back at 5pm UK with the solutions to the questions, the results of the survey and a discussion.

SPOILER POLICY: If you think that the answers to all the questions above are the same, feel free to argue your case. But if you get different answers keep your workings to yourself and we’ll discuss them from 5pm.

The discussion might get quite heated. When it comes to the two child problem, it often does. Since it was first stated in the late 1950s, it has regularly provoked frenzies of disbelief and anger. You have been warned.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

My new book So You Think You’ve Got Problems? is a compendium of 200 or so puzzles with the stories behind them. One of the problems is the one discussed in today’s column, with added material about the history and development of probability ‘paradoxes’ from the nineteenth century to the present day. The other puzzles span wordplay, logic, geometry, linguistics, topology and many other fields. It’s for all abilities: there are very simple teasers that children will be able to do and utterly baffling ones for only the sharpest minds.