Question

A man tosses an unbiased coin $3$ times. If he gets a head all the $3$ times he is to get a prize of Rs. $160$ along with his entry fee of Rs. $16$, otherwise, he loses his entry fee. The mathematical expectation of his profit is

• A. $20$
• B. $14$
• C. $6$
• D. $12$

Answer 1

Answer: (C)

$6$

Expectation is given by $E\left[X\right]=\sum p_i{x}_i$
where $x_i$ are the values variable $X$ can take and $p_i$ is the probability of occurance of $x_i$
Now,

$P($win$)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}$
And value gained on winning $=160$

$P($loss$)=1-\frac{1}{8}=\frac{7}{8}$
Value gained on losing $=-16$

(Entry fee is lost, therefore negative gain)
$\Rightarrow$ Expected value $=\frac{1}{8}\times 160+\frac{7}{8}\times (-16)=6$

Hence, option C.