Question

A person plays a game of tossing a coin thrice. For each head, he is given Rs $2$ by the organiser of the game and for each tail, he has to give Rs $1.50$ to the organiser. Let $X$ denotes the amount gained or lost by the person. Then range of $X$ is:
(Where minus sign shows the loss to the player and positive sign shows gain to the player)

• A. $\{-1\}$
• B. $\{-1,2.5\}$
• C. $\{-4.5,-1,2.5\}$
• D. $\{-4.5,-1,2.5,6\}$

Answer 1

The correct option is D $\{-4.5,-1,2.5,6\}$

$X$ is a number whose values are defined on the outcomes of a random experiment. Therefore, $X$ is a random variable.
Now, sample space of the experiment is:
$S=\{HHH,THH,HTH,HHT,TTT,HTT,THT,TTH\}$
Then $X\left(HHH\right)=Rs(2\times 3)=Rs6$
$X\left(HHT\right)=X\left(HTH\right)=X\left(THH\right)=Rs(2\times 2-1\times 1.50)=Rs2.50$
$X\left(HTT\right)=X\left(THT\right)=X\left(TTH\right)=Rs(1\times 2)-(2\times 1.50)=-Rs1$
and $X\left(TTT\right)=-Rs(3\times 1.50)=-Rs4.50$
Where, minus sign shows the loss to the player. Thus, for each element of the sample space, $X$ takes a unique value. Hence, $X$ is a function on the sample space whose range is
$\{-4.50,-1,2.50,6\}$