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Math Help - Maths help! Desperate!

  1. #1
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    Maths help! Desperate!

    Take a look at the photo
    May I have the method aswell for future reference?
    Many thanks!
    Attached Thumbnails Attached Thumbnails Maths help! Desperate!-image.jpg  
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  2. #2
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    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.

    Maths help! Desperate!-clipboard01.jpg

    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

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