Question

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:

• A. 51
• B. 48
• C. 32
• D. 64

Hint:

To calculate the expected value and variance for a probability distribution, use the formulas for mean and variance, and then apply them to the given probabilities.

Answer 1

The Correct Option is B

For each possible outcome: HHH maps to 0, HHT maps to 0, HTH maps to 1, HTT maps to 0, THH maps to 1, THT maps to 1, TTH maps to 1, TTT maps to 0. Probability distribution: \( \mu = \sum x_i P_i = \frac{1}{2} \) \( \sigma^2 = \sum x_i^2 P_i - \mu^2 = \frac{1}{2} \times 1^2 + \frac{1}{2} \times 1^2 - \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) \( 64(\mu + \sigma^2) = 64\left(\frac{1}{2} + \frac{1}{4}\right) = 64 \times \frac{3}{4} = 48 \]