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

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