In a game, a man wins Rs.
Question:

In a game, a man wins Rs. 100 if he gets 5 of 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 :

1. $\frac{400}{3}$ gain

2. $\frac{400}{3} \operatorname{loss}$

3. 0

4. $\frac{400}{9}$ loss

Correct Option: , 3

Solution:

Expected Gain/ Loss $=$

$=\mathrm{w} \times 100+\mathrm{Lw}(-50+100)+\mathrm{L}^{2} \mathrm{w}(-50-50+$

$100)+\mathrm{L}^{3}(-150)$

$=\frac{1}{3} \times 100+\frac{2}{3} \cdot \frac{1}{3}(50)+\left(\frac{2}{3}\right)^{2}\left(\frac{1}{3}\right)(0)+$

$\left(\frac{2}{3}\right)^{3}(-150)=0$

here $\mathrm{L}$ denotes probability that outcome

$1,2,3,4\left(\mathrm{~L}=\frac{4}{6}=\frac{2}{3}\right)$