Question

One purse contains $1$ sovereign and $3$ shillings, a second purse contains $2$ sovereigns and $4$ shillings, and a third contains $3$ sovereigns and $1$ shilling. If a coin is taken out of one of the purses selected at random, find the chance that it is a sovereign.

Answer 1

Since each purse is equally likely to be taken, the chance of selecting the first is $\frac{1}{3};$ and the chance of then drawing a sovereign is $\frac{1}{4};$ hence the chance of drawing a sovereign so for as it depends upon the first purse is $\frac{1}{3}\times \frac{1}{4},$ or $\frac{1}{12}$. Similarly the chance of drawing a sovereign so far as it depends on the second purse is $\frac{1}{3}\times \frac{2}{6}$, or $\frac{1}{9}$; and from the third purse the chance of drawing a sovereign is $\frac{1}{3}\times \frac{3}{4}$, or $\frac{1}{4}$;
$\therefore$ The required chance $=\frac{1}{12}+\frac{1}{9}+\frac{1}{4}=\frac{4}{9}$.