Question

From a bag containing $2$ sovereigns and $3$ shillings a person is allowed to draw $2$ coins indiscriminately; find the value of his expectation.

Answer 1

$1$ soveriegn $=20$ shillings

$2$ coins from $5$ coins can be drawn in ${}^5{C}_2=10$ ways

If both are sovereign then expected value $=\frac{{}^2{C}_2}{{}^5{C}_2}\times 40=4$

If both are shillings then expected value $=\frac{{}^3{C}_2}{{}^5{C}_2}\times 2=\frac{3}{5}$

If one is shilling and one is sovereign then expected value $=\frac{{}^3{C}_1{\times }^2{C}_1}{{}^5{C}_2}\times 21=\frac{63}{5}$

Total expected value $=4+\frac{3}{5}+\frac{63}{5}=\frac{86}{5}$