Question

A purse contains five coins, each of which may be a shilling or a six pence; two are drawn and found to be shillings: find the probable value of the remaining coins.

Answer 1

We know that

The chance of $5$ shillings is $\frac{1}{2^5}$

$4$ shillings is $\frac{5}{2^5}$

$3$ shillings is $\frac{10}{2^5}$

$2$ shillings is $\frac{10}{2^5}$

$P_1=\frac{1}{32},P_2=\frac{5}{32},P_3=\frac{10}{32},P_4=\frac{10}{32},$

Also $P_1=1,P_2=\frac{4}{5}\times \frac{3}{4},P_3=\frac{3}{5}\times \frac{2}{4},P_4=\frac{2}{5}\times \frac{1}{4},$

$\therefore P_1{P}_1=\frac{1}{32},P_2{P}_2=\frac{3}{32},P_3{P}_3=\frac{3}{32},P_4{P}_4=\frac{1}{32}$

$\therefore \frac{Q_1}{1}=\frac{Q_2}{3}=\frac{Q_3}{3}=\frac{Q_4}{1}=\frac{1}{8}$

Hence the probable value in shillings of the remaining coins

$=\frac{1}{8}\times 3+\frac{3}{8}\times \frac{5}{2}+\frac{3}{8}\times 2+\frac{1}{8}\times \frac{3}{2}=\frac{9}{4}=2\frac{1}{4}$ shillings