# The Two-Ewes-Day Problem

## How sheep rescued a classic puzzle

A few years ago I wrote a blog about the famous and much debated 'boy-girl problem', and a related puzzle known as the 'Tuesday boy problem' (you can read the blog here ).

As a reminder, the basic puzzle is this:

I have two children, at least one of whom is a boy.  What is the chance that the second child is also a boy?

[SPOILER: THE SOLUTION IS DISCUSSED BELOW.  You might want to think about the puzzle first.  If you haven't seen this week's puzzle in New Scientist, take a look:  https://twitter.com/robeastaway/status/1393553018893881349]

The classic, surprising answer is that the chance of a second boy is 1/3, not 1/2 as most would expect.  Why?  Because if I have two children, there is an equal chance that they will be Boy-Boy, Boy-Girl, Girl-Boy or Girl-Girl.  There are three equally likely combinations in which at least one child is a boy (BB, BG and GB) and only in the first scenario is the other child also a boy.

However - as I argued in my old blog - this answer all depends on how the information about having two children is revealed, and I could think of no 'natural' situation in which you would happen to spontaneously discover somebody had two children at least one of whom was a boy.

That was until I heard from US mathematician Peter Winkler a couple of months ago.  He told me of a friend who was pregnant with 'fraternal' (ie non-identical) twins. As part of the pregnancy monitoring, the mother had undergone an extremely reliable test which always detects Y-chromosomes if at least one child is a boy.  The test for Y-chromosomes was positive, so the mother was told that at least one of the twins was a boy. She hoped the other might be a girl, but thought the odds were no better than 50-50.

Because fraternal twins are the same as two independently born children, the sex of each child is a 50:50 random choice, like flipping a coin. And so, in this instance, knowing that one child was a boy, the chance that the other would be a girl was 2/3.  As it turns out, the other baby was indeed a girl, so the mum got what she wanted.

AND THEN CAME THE SHEEP

Zoe Mensch and I thought this would make a nice puzzle for New Scientist, especially with the scientific testing element. However, at a time when binary categorisation into boys/girls is a topic of heated public and scientific debate (see the footnote for an interesting Scientific American article about this), we decided it would be better if the puzzle could be about animals rather than humans.

After a couple of hours of research, I landed on sheep as the perfect solution.  Typically sheep arrive as non-identical twins, and like most mammals, they are roughly 50-50 male/female.  I even checked with the world famous Roslin Institute (remember Dolly the Sheep?), to confirm that a Y chromosome test on sheep could identify that at least one twin was a male, and that intersex sheep are almost unheard of.

We gave the puzzle a story.  A rare-breed sheep farmer's prize ewe is expecting twins, at least one of which he knows will be male.  He'd love the other to be female so that there will be two ewes, mother and daughter, on the farm next year, but what are the chances?

And that, dear reader, is how this new version of the puzzle came to be called The Two-Ewes-Day Problem.

* You can read about the extraordinary complexity of sex determination here:

https://www.scientificamerican.com/article/beyond-xx-and-xy-the-extraordinary-complexity-of-sex-determination/