Question

A purse contains $10$ ten rupee coins and $5$ five rupee coins. Two coins are randomly drawn. If the expected value of $2$ drawn coins is $\lambda ,$ then $3\lambda $ is equal to

Answer 1

$2$ coins can be both $10$ rupee coins, both $5$ rupee coins or one $5$ rupee coin and one $10$ rupee coin
So, expected value
$=\frac{^{10} C_{2}}{^{15} C_{2}}\times 20+\frac{^{5} C_{2}}{^{15} C_{2}}\times 10+\frac{^{10} C_{1}^{5} C_{1}}{^{15} C_{2}}\times 15$ $=\frac{3}{7}\times 20+\frac{2}{21}\times 10+\frac{10}{21}\times 15$
$=\frac{350}{21}=\frac{50}{3}$
i.e. $\lambda =\frac{50}{3}$
Hence, $3\lambda =50$