A coin is tossed three times. Let X denote the number of times a tail follows a head. If and denote the mean and variance of X, then the value of is L [2025]

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Solution

Q.
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 : [2025]

1 64

2 48

3 51

4 32

Ans.
(2)

When a coin is tossed three times, outcomes are:

{HHH, HHT, HTH, THH, THT, TTH, HTT, and TTT}

Outcomes when tails does not follows a head (only once):

{HHH, THH, TTT, TTH}

Outcomes when tail follows a head:

{HHT, HTT, HTH, THT}

$∴$ Probability distribution table is given by

$X_{i}$ 0 1

$P (X_{i})$ 1/2 1/2

Mean $(μ) = \sum_{}^{} X_{i}^{} P_{i} = \frac{1}{2}$

and variance $(σ^{2}) = \sum_{}^{} X_{i}^{2} P_{i} - μ^{2} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$

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

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