Question

A player tosses $2$ fair coins. He wins Rs$5$ if $2$ heads appear, Rs$2$ if head appears and Rs$1$ if no head appears. Find his expected winning amount and variance of winning amount.

Answer 1

No. of coin $=2$

$\Rightarrow$ Sample space $=2^2=4=[HH,HT,TH,TT]$

$\Rightarrow$ Probability of getting $2$ heads, i.e, $P_1=\frac{1}{4}$ $\therefore n_1=Rs.5$

Probability of getting $1$ head, i.e, $P_1=\frac{2}{4}$ $\therefore n_2=Rs.2$

Probability of getting no head, i.e, $P_3=\frac{1}{4}$ $\therefore n_3=Rs.1$

$\Rightarrow$ Expectation $=n_1{p}_1+n_2{p}_2+n_3{p}_3$

$=\frac{5}{4}+\frac{2\times 2}{4}+\frac{1}{4}$

$=\frac{5+4+1}{4}$

$=\frac{10}{4}$

$=\frac{5}{2}$

Hence, the answer is $\frac{5}{2}.$