A boy tosses a fair coin 3 times. If he gets | vedclass

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(C) Let $X$ be the random variable representing the number of heads in $3$ tosses of a fair coin. The possible values of $X$ are $0, 1, 2, 3$.The prob

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$3$

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(C) Let $X$ be the random variable representing the number of heads in $3$ tosses of a fair coin. The possible values of $X$ are $0, 1, 2, 3$.The probability distribution of $X$ is given by the binomial distribution $B(n=3, p=0.5)$:$P(X=0) = \binom{3}{0} (0.5)^3 = \frac{1}{8}$$P(X=1) = \binom{3}{1} (0.5)^3 = \frac{3}{8}$$P(X=2) = \binom{3}{2} (0.5)^3 = \frac{3}{8}$$P(X=3) = \binom{3}{3} (0.5)^3 = \frac{1}{8}$Let $Y$ be the gain,where $Y = 2X$. The expected gain $E[Y]$ is given by $E[2X] = 2E[X]$.For a binomial distribution,$E[X] = np = 3 	imes 0.5 = 1.5$.Therefore,$E[Y] = 2 	imes 1.5 = 3$.Alternatively,using the table:$X$$0, 1, 2, 3$$Y = 2X$$0, 2, 4, 6$$P(Y)$$\frac{1}{8}, \frac{3}{8}, \frac{3}{8}, \frac{1}{8}$$E[Y] = 0(\frac{1}{8}) + 2(\frac{3}{8}) + 4(\frac{3}{8}) + 6(\frac{1}{8}) = \frac{0+6+12+6}{8} = \frac{24}{8} = 3$.

$X = x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$P(X = x)$	$\frac{1}{10}$	$k$	$\frac{1}{5}$	$2k$	$\frac{3}{10}$	$k$

$A$ boy tosses a fair coin $3$ times. If he gets $₹ 2x$ for $x$ heads,then his expected gain equals to $₹........$

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