Question

In a purse there are $5$ sovereigns and $4$ shillings. If they are drawn out one by one, what is the chance that they come out sovereigns and shillings alternately, beginning with a sovereign?

Answer 1

Probability of drawing a sovereign out of $5$ sovereigns and $4$ shillings is $\frac{5}{9}$

Probability of drawing a shilling out of $4$ sovereigns and $4$ shillings is $\frac{4}{8}$

Probability of drawing a sovereign out of $4$ sovereigns and $3$ shillings is $\frac{4}{7}$

Probability of drawing a shilling out of $3$ sovereigns and $3$ shillings is $\frac{3}{6}$

Probability of drawing a sovereign out of $3$ sovereigns and $2$ shillings is $\frac{3}{5}$

Probability of drawing a shilling out of $2$ sovereigns and $2$ shillings is $\frac{2}{4}$

Probability of drawing a sovereign out of $2$ sovereigns and $1$ shillings is $\frac{2}{3}$

Probability of drawing a shilling out of $1$ sovereigns and $1$ shillings is $\frac{1}{2}$

Probability of drawing a sovereign out of $1$ sovereigns and $0$ shillings is $1$

Hence, total probability is $\frac{5}{9}\times \frac{4}{8}\times \frac{4}{7}\times \frac{3}{6}\times \frac{3}{5}\times \frac{2}{4}\times \frac{2}{3}\times \frac{1}{2}\times 1=\frac{5!\times 4!}{9!}=\frac{1}{126}$