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 $64\left(\mu+\sigma^{2}\right)$ is :

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Probability

A coin is tossed three times. Let X denote the number of times a tail follows a head. If

μ

and

σ^{2}

denote the mean and variance of X, then the value of

64 (μ + σ^{2})

is :

Answer: (B)

Toss a coin three times and let

X

be the number of times an H is immediately followed by a T. Listing all 8 equally likely outcomes:

HHH: no H

\to

T pair,

X = 0

HHT: position 2–3 is H

\to

X = 1

HTH: position 1–2 is H

\to

X = 1

HTT: position 1–2 is H

\to

X = 1

THH: no H

\to

T pair,

X = 0

THT: position 2–3 is H

\to

X = 1

TTH: no H

\to

T pair,

X = 0

TTT: no H

\to

T pair,

X = 0

X

takes value 1 in exactly 4 of the 8 outcomes and value 0 in the other 4:

P (X = 0) = \frac{1}{2}, P (X = 1) = \frac{1}{2} .

Mean:

μ = 0 \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} = \frac{1}{2}

.

Variance:

E (X^{2}) = 0 \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} = \frac{1}{2}

, so

σ^{2} = E (X^{2}) - μ^{2} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}

64 (μ + σ^{2}) = 64 (\frac{1}{2} + \frac{1}{4}) = 64 \cdot \frac{3}{4} = 48.

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option