Question

A and B draw from a purse containing $3$ sovereigns and $4$ shillings: find their respective chances of first drawing a sovereign, the coins when drawn not being replaced.

Answer 1

Probability b that $A$ draws a sovereign $=\frac{3}{7}$

Probability that $A$ fails and $B$ draws $=\frac{4}{7}.\frac{3}{6}=\frac{4}{7}.\frac{1}{2}=\frac{2}{7}$

Probability that $A$ and $B$ both fails then $A$ draws a sovereign $=\frac{4}{7}.\frac{1}{2}.\frac{3}{5}=\frac{6}{35}$

Probability that $A$ fails twice and then $B$ draws a sovereign $=\frac{4}{7}.\frac{1}{2}.\frac{3}{5}.\frac{2}{4}=\frac{3}{35}$

Probability that when both $A$ and $B$ fails twice then $A$ draws a sovereign $=\frac{4}{7}.\frac{1}{2}.\frac{3}{5}.\frac{2}{4}.\frac{2}{3}=\frac{2}{35}$

$\therefore$ chance of winning of $A$ $=\frac{3}{7}+\frac{6}{35}+\frac{2}{35}=\frac{23}{35}$

and chance of winning of $B$ $=\frac{2}{7}+\frac{3}{35}=\frac{13}{35}$