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

Answer 1

Answer: (B)

48

Answer 2

For three coin tosses, label the tosses as T₁, T₂, T₃. A “tail follows a head” is counted when a head occurs immediately before a tail (i.e., in pairs (T₁,T₂) and (T₂,T₃)). Listing all 8 outcomes shows:

• Outcomes with X = 1: HHT, HTH, HTT, THT (4 outcomes)
• Outcomes with X = 0: HHH, THH, TTH, TTT (4 outcomes)

Thus,

$$\mu = E[X] = \frac{0 \cdot 4 + 1 \cdot 4}{8} = 0.5,$$ $$E[X^2] = \frac{0^2 \cdot 4 + 1^2 \cdot 4}{8} = 0.5,$$ $$\sigma^2 = E[X^2] - (E[X])^2 = 0.5 - (0.5)^2 = 0.25.$$

Then,

$$64(\mu + \sigma^2) = 64(0.5 + 0.25) = 64 \times 0.75 = 48.$$