Question

In a game, a man wins Rs. $100$ if he gets $5$ or $6$ on a throw of a fair die and loses Rs. $50$ for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

• A. $\frac{400}{9}$ loss
• B. $\frac{400}{3}$ loss
• C. $\frac{400}{3}$ gain
• D. $0$

Answer 1

The correct option is D $0$

$A:$ Event that $5$ or $6$ appear on die.
$B:$ Event that any other number appear on die.
Therefore, following events are possible.
$A,BA,BBA,BBB$
$X:$ Random variable denoting the amount won.
$\begin{array}{ccccc}\text{X}& +100& +50& 0& -150\\ \text{P(X)}& \frac{1}{3}& \frac{2}{9}& \frac{4}{27}& \frac{8}{27}\end{array}$

$E\left(X\right)=\frac{100}{3}+\frac{100}{9}+0-\frac{1200}{27}=0$