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(B) The sample space for tossing a coin three times is $S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$.Let $X$ be the number of times a tail follows

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$48$

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(B) The sample space for tossing a coin three times is $S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$.Let $X$ be the number of times a tail follows a head (i.e., the pattern $HT$).$HHH ightarrow 0$$HHT ightarrow 1$ (tail follows head at the end)$HTH ightarrow 1$ (tail follows head at the start)$HTT ightarrow 1$ (tail follows head at the start)$THH ightarrow 0$$THT ightarrow 1$ (tail follows head at the end)$TTH ightarrow 0$$TTT ightarrow 0$Wait, let us re-evaluate the occurrences of $HT$:$HHH: 0$$HHT: 1$ ($HT$ at index $1-2$)$HTH: 1$ ($HT$ at index $1-2$)$HTT: 1$ ($HT$ at index $1-2$)$THH: 0$$THT: 1$ ($HT$ at index $2-3$)$TTH: 0$$TTT: 0$The values of $X$ are $0$ and $1$.$P(X=0) = \frac{4}{8} = \frac{1}{2}$$P(X=1) = \frac{4}{8} = \frac{1}{2}$Mean $\mu = E[X] = 0 	imes \frac{1}{2} + 1 	imes \frac{1}{2} = \frac{1}{2}$.Variance $\sigma^2 = E[X^2] - (E[X])^2 = (0^2 	imes \frac{1}{2} + 1^2 	imes \frac{1}{2}) - (\frac{1}{2})^2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$.The value of $64(\mu + \sigma^2) = 64(\frac{1}{2} + \frac{1}{4}) = 64(\frac{3}{4}) = 48$.

$A$ coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of $X$, then the value of $64(\mu+\sigma^2)$ is :

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