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:
Step-by-Step Solution
Key Concept: List all $2^3 = 8$ outcomes and identify the value of $X$ (number of TH patterns) for each. Build the probability distribution and compute $\mu$ and $\sigma^2$.
Distribution: $X=0$ for HHH, TTT, THH, TTH with prob 4/8; $X=1$ for HHT, HTH, THT, HTT with prob 4/8. $\mu = 0 \cdot \frac{4}{8} + 1 \cdot \frac{4}{8} = \frac{1}{2}$. $\sigma^2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$. $64(\mu + \sigma^2) = 64 \times \frac{3}{4} = 48$.
Correct Answer: 48