A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ² denote the mean and variance of X, then the value of 64(μ+σ²) is:

Solution

Maths sir · 3 steps

List all 8 outcomes and count the HT pairs. You will see that X can only be 0 or 1.

1. Where HT can occur. With three tosses there are only two neighbouring pairs: tosses 1 and 2, and tosses 2 and 3. Two HT pairs would need H T H T, which takes four tosses, so X is 0 or 1.

   X	Outcomes	Probability
   0	HHH, THH, TTH, TTT	4/8 = 1/2
   1	HHT, HTH, HTT, THT	4/8 = 1/2

2. Mean and variance. μ = 0 × 1/2 + 1 × 1/2 = 1/2. Also E(X²) = 1/2, so σ² = E(X²) - μ² = 1/2 - 1/4 = 1/4.

   64(μ + σ²) = 64(1/2 + 1/4) = 64 × 3/4 = 48

3. So. The value is 48, which is option (d).

The idea to keep. When a random variable counts a pattern, first ask which values it can take. Then list the outcomes, and always use σ² = E(X²) - μ².

Answer from the NTA final answer key. Solution by PrepWiser.