# Maths help! Desperate!

• Feb 23rd 2014, 10:35 AM
Bentheman14
Maths help! Desperate!
Take a look at the photo :)
May I have the method aswell for future reference?
Many thanks!
• Feb 23rd 2014, 01:22 PM
romsek
Re: Maths help! Desperate!
Ok what you do here is construct a probability tree for a single couple. Each couple probabilistically behaves the same so the overall ratio for the population is just the same as that for a single couple

Note that this all assumes the children don't themselves start forming couples (from other families hopefully) and themselves reproducing.

Attachment 30230

if you work through the thing you'll find that

$$Pr[\mbox{k girls, 1 boy}]=\left(\frac{1}{2}\right)^{k+1}$$

thus

$$P[\mbox{n children}]=\left(\frac{1}{2}\right)^{n}$$

For n children the ratio of girls to boys is

$$r[n]=\frac{1}{n-1}$$

and so you end up with a discrete probability distribution for certain ratios, i.e.

$$Pr\left[r[n]\right]=\left(\frac{1}{2}\right)^{n}$$

and zero for all other numbers. To make things a bit more concrete

\begin{align*} &Pr[\infty]=\frac{1}{2} \\ \\ &Pr[1]= \frac{1}{4} \\ \\ &Pr\left[\frac{1}{2}\right]=\frac{1}{8} \\ \\ &\mbox{and in general for n>1} \\ \\ &Pr\left[\frac{1}{n-1}\right]=\frac{1}{2^n} \end{align*}