# Bayes’ Theorem, summer babies and that funny | symbol

There's an excellent article by @johnallenpaulos talking about a really interesting probability 'paradox' to do with summer births.

It's not really a paradox (as with a lot of probability, it makes perfect sense once you think it through), but the "at least one boy" puzzle was one of my favourites growing up.

## An excuse to talk about Bayes's Theorem!

What Paulos's article doesn't do, though, is provide a nice, colourful table to illustrate either of the puzzles. In fairness, the seasons one would be a bit big, but the simple version? Let me put that right.

If we know 100 families with two children, we'd expect their distribution to look something like this[1]:

Older boy Older girl 25 25 25 25

That is, if you want to know the probability of the family being made up like mine - I have a younger brother - you look down the older boy column until you reach the younger boy row; there were 25 families that matched that description, out of 100 altogether. That's 25/100 = 25% or one in four.

Paulos's first example asks about the probability of the second child being a boy if you know the first child is a boy. "If you know..." is usually translated into maths as "Given" - which is something that comes up a lot in S1, and makes everyone cry (unless they're up to speed with Bayes' Theorem). You might even see it as $P(two~boys|older~boy)$ - which you read as "the probability of 'two boys' given 'older boy'".

I'll try to wipe those tears away with a Really Clear Table.

Older boy Older girl 25 25 25 25

We're only interested (here) in families with an older boy. (Whenever you've got a "given" question, you want to highlight the events AFTER the | symbol - they're the ones you know are possible).

We see there are 50 families in total there, and 25 of them have a second boy - the probability is $\frac{25}{50} = \frac{1}{2}$, as you'd expect.

The tricky bit comes in Paulos's second example, where you know at least one of the children is a boy - but not which one. Here, we're looking for $P(two~boys|at~least~one~boy)$ - and we want to highlight every family with at least one boy.

Older boy Older girl 25 25 25 25

Now it's quite clear, I hope: there are 75 families with at least one boy; 25 of them have two boys. The probability of two boys given at least one is $\frac{25}{75} = \frac{1}{3}$.

The really clear table method works in almost all cases where you could use a Venn diagram - the only times you actually need a Venn diagram are when the question asks for it, or if there are more than two events.

## The really odd thing about the summer babies problem

Imagine you ask someone about their kids and they say "I've got two kids -- actually, I'm just off to pick my son up". From the information you have, you'd say it was a $\frac{2}{3}$ chance that the other child was a daughter.

If instead they said "I've got two kids -- actually, I'm just off to pick my son up from his birthday party," the probability changes to a little over $\frac{1}{2}$ -- even though the parent has said nothing about the other child.

Bayes' Theorem has a lot to answer for. What's the probability there's some ibuprofen in the bathroom?

[1] In real life, you'd also expect it to vary a bit - so although this is the most likely distribution, the chances of it being exactly this are actually very small - that's a story for another day.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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