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 1: List Sample Space and Values of X

First, we list all possible outcomes when a coin is tossed three times. There are $2^3 = 8$ possible outcomes. For each outcome, we determine the value of $X$, which is the number of times a tail follows a head (HT sequence).

Step 2: Determine Probability Distribution of X

From the sample space, we count the occurrences for each value of $X$: $X=0$ (HHH), $X=1$ (HHT, THH, THT, TTH), $X=2$ (HTH, HTT), $X=3$ (none). Wait, let's re-evaluate the counts for X. HHH has 0 HT. HHT has 1 HT. HTH has 1 HT. THH has 0 HT. HTT has 1 HT. THT has 1 HT. TTH has 0 HT. TTT has 0 HT. So, $X=0$ for HHH, THH, TTH, TTT (4 outcomes). $X=1$ for HHT, HTH, HTT, THT (4 outcomes). $X=2$ and $X=3$ have 0 outcomes. This means the problem statement or my interpretation of 'tail follows a head' might be slightly off. Let's re-read carefully: 'number of times a tail follows a head'. Let's re-evaluate $X$ for each outcome: HHH: 0 (no HT) HHT: 1 (H_H_T) HTH: 1 (H_T_H) THH: 0 (no HT) HTT: 1 (H_T_T) THT: 1 (T_H_T) TTH: 0 (no HT) TTT: 0 (no HT) So, $X=0$ for HHH, THH, TTH, TTT (4 outcomes). $X=1$ for HHT, HTH, HTT, THT (4 outcomes). This means $P(X=0) = 4/8 = 1/2$ and $P(X=1) = 4/8 = 1/2$. All other probabilities are 0. Let's correct the table.

Step 3: Calculate Mean (μ)

The mean, $\mu$, also known as the expected value $E(X)$, is calculated by summing the product of each possible value of $X$ and its corresponding probability.

Step 4: Calculate Variance (σ²)

The variance, $\sigma^2$, is calculated using the formula $E(X^2) - (E(X))^2$. First, we find $E(X^2)$ by summing the product of the square of each $X$ value and its probability. Then, we subtract the square of the mean.

Step 5: Calculate the Final Expression

Finally, we substitute the calculated values of the mean ($\mu$) and variance ($\sigma^2$) into the given expression $64(\mu + \sigma^2)$ and simplify to find the final answer.

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