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In a game of throwing 3 coins, a player will loose ₹ $5 /$ - for each head and gain ₹ 10 - for each tail. If a random variable $X: S \rightarrow R$ is defined as $X(a)=$ net gain $(a \in S)$, then the mean of the random variable is (in rupees)
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Verified Answer
The correct answer is:
$\frac{15}{2}$
If 3 coins are throw together, then $S=\{H H H, H H T, H T H, H T, T H H, T H T, T H, T T T\}$ Then, $x$ has value $-15(H H H), 0(H H T, H T H, T H H)$, $15(\mathrm{HTT}, \mathrm{THT}, \mathrm{TTH})$ and $30(\mathrm{MT})$.
$\therefore$ Probability distribution is given by
\begin{array}{ccccc}
\hline \boldsymbol{x} & -15 & 0 & 15 & 30 \\
\hline \boldsymbol{P}(\boldsymbol{x}) & \frac{1}{8} & \frac{3}{8} & \frac{3}{8} & \frac{1}{8}
\end{array}
$$
\begin{aligned}
& \therefore \quad \text { Mean }=\Sigma \chi(P(x) \\
&=\frac{-15}{8}+\frac{0}{8}+\frac{45}{8}+\frac{30}{8}=\frac{60}{8}=\frac{15}{2}
\end{aligned}
$$
$\therefore$ Probability distribution is given by
\begin{array}{ccccc}
\hline \boldsymbol{x} & -15 & 0 & 15 & 30 \\
\hline \boldsymbol{P}(\boldsymbol{x}) & \frac{1}{8} & \frac{3}{8} & \frac{3}{8} & \frac{1}{8}
\end{array}
$$
\begin{aligned}
& \therefore \quad \text { Mean }=\Sigma \chi(P(x) \\
&=\frac{-15}{8}+\frac{0}{8}+\frac{45}{8}+\frac{30}{8}=\frac{60}{8}=\frac{15}{2}
\end{aligned}
$$
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