## Problem Children

The title is perhaps a little misleading. This post will by no account help you deal with problem children (unless they enjoy quirky maths blogs, in which case you’re welcome). That is not the purpose of this blog, no sir. Very few issues involving problem children have mathematical solutions and that is something that just doesn’t sit right with me.

So now we know what this post is not about, I suppose it’s time to get cracking. The idea I’m covering today is conditional probability; what is the chance of one event given that another event has already occurred (more formally P(A|B)). For instance, while the probability of missing a leg or being a lion tamer may, individually, be quite low, the probability of missing a leg given that a person is a lion tamer is considerably higher (as well as an increased probability that everyone around them wishes they were lion tamers).

And you thought your first kiss was scary

Let me paint a scene for you. It’s a beautiful, warm night. The stars are out and the moon illuminates the night sky. The sound of music and laughter fills the air and you don’t notice any of this because you’re in the library idly browsing through the latest mathematical journals. It’s a typical Friday night (isn’t it?).

All of a sudden, the librarian appears out of nowhere and begins sorting some of the books nearby. After a few minutes in close proximity you decide to break the tension and talk. After exchanging pleasantries (“nice night, isn’t it?”) and a few arkwardries (“those glasses frame your face really well”) she mentions that she has two children, at least one of which is a boy. On this note she departs to organise another section, leaving you to ponder the implications of the conversation.

Unable to comprehend why she has been so cryptic about the gender of her children (a flirting technique, maybe?), you sit down and flick through a copy of ‘Macho Maths Weekly’ to calm your nerves.

So here’s the question: given that one of her children is a boy, what it the probability her other child is also a boy? All seems very simple, doesn’t it? If there’s a 50/50 chance a child is a boy or a girl, then the probability must be $\frac{1}{2}$. Thought I’d try and bluff you, didn’t I? All this talk of conditional probability used to distract you from the obvious answer. But as is usually the case with this blog, things aren’t quite that simple.

Pictured: the house of math

To work this out, we need to consider the different possibilities. With two children we have 4 combinations –

Girl Girl

Girl Boy

Boy Girl

Boy Boy

Since we’re given that one of them is a boy (ruling out ‘Girl Girl’), we are left with 3 equally likely possibilities of which only one includes another boy. we therefore conclude that the probability the other is a boy is $\frac{1}{3}$. Bet you didn’t see that coming.

What if she tells you she has a boy born on a Tuesday (ok, I really don’t know why she would do this, just run with it)? Surely it’s $\frac{1}{3}$ again. How can the day he’s born have any effect on whether the other other child is a boy? Let’s have a look.

We have a few more combinations here, 27 to be precise. Again any possibilities not involving a boy born on tuesday are eliminated leaving –

Boy (Tues) Girl (Any) – 7 combinations (Girl (Mon), Girl (Tues), etc)

Girl (Any) Boy (Tues) – 7 combinations

Boy (Tues) Boy (Any) – 7 combinations

Boy (Any) Boy (Tues) – 6 combinations (As Boy (Tues) Boy (Tues) has already been accounted for)

We therefore have 27 equally likely outcomes, 13 of which involve another boy. The probability is therefore $\frac{13}{26}$! Oddly enough, the more detail she gives about her son, the closer the probability that her other child is a boy gets to ﻿$\frac{1}{2}$.

Who wouldn't want two boys?

Have children? Want to amaze your friends with simple probability tricks? Then please refrain from using the sentence “at least one of my children is a boy”. Only Social Services will take interest in statements like that.

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### 8 Comments on “Problem Children”

1. Deborah Says:

This is very funny and very informative. It provided some light relief during a break from painting an intransigent radiator. What’s the probability that my paint job will turn out OK in the end?x

• mostlymath Says:

Glad you enjoyed it. I’d say fairly close to one, lot of faith in your painting ability. Glad my post compared well to watching paint dry.

2. […] says conditional probability is hard? James Pollock explains it like one plus one in his post Problem Children in A mathematicians guide to… Believe me, this guy is one heck of a […]

3. Sally Says:

Get cracking on another post, lad!

4. Gordon Says: