Probability 1 Question 1
1. A person throws two fair dice. He wins ₹ 15 for throwing a doublet (same numbers on the two dice), wins ₹ 12 when the throw results in the sum of 9 , and loses ₹ 6 for any other outcome on the throw. Then, the expected gain/loss (in ₹) of the person is
(a) $\frac{1}{2}$ gain
(b) $\frac{1}{4}$ loss
(c) $\frac{1}{2} \operatorname{loss}$
(d) 2 gain
Show Answer
Answer:
Correct Answer: 1. (c)
Solution:
- It is given that a person wins ₹15 for throwing a doublet $(1,1)(2,2),(3,3)$, $(4,4),(5,5),(6,6)$ and win ₹ 12 when the throw results in sum of 9 , i.e., when $(3,6),(4,5)$,
$(5,4),(6,3)$ occurs.
Also, losses ₹ 6 for throwing any other outcome, i.e., when any of the rest $36-6-4=26$ outcomes occurs.
Now, the expected gain/loss
$=15 \times P$ (getting a doublet) $+12 \times P$ (getting sum 9) $-6 \times P$ (getting any of rest 26 outcome)
$$ \begin{aligned} & =15 \times \frac{6}{36}+12 \times \frac{4}{36}-6 \times \frac{26}{36} \\ & =\frac{5}{2}+\frac{4}{3}-\frac{26}{6}=\frac{15+8-26}{6} \\ & =\frac{23-26}{6}=-\frac{3}{6}=-\frac{1}{2}, \text { means loss of } ₹ \frac{1}{2} \end{aligned} $$
