Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(x = 4), then the sum of the mean and the variance of X is :

Solution

Maths sir · 3 steps

Write both probabilities with the binomial formula: the coefficients match, so the ratio gives p at once.

1. Write both terms. For B(7, p) with q = 1 - p, P(X = r) = C(7, r) pʳ q⁷⁻ʳ. So P(X = 3) = C(7, 3) p³ q⁴ and P(X = 4) = C(7, 4) p⁴ q³.
2. Cancel and solve. Both coefficients are 35, since C(7, 3) = C(7, 4). Cancelling 35p³q³ from both sides of P(X=3) = 5P(X=4) leaves q = 5p. With p + q = 1, this gives 6p = 1, so p = 1/6 and q = 5/6.

   np + npq = 7 × 1/6 + 7 × 1/6 × 5/6 = 42/36 + 35/36

3. So. The mean is 7/6 and the variance is 35/36. Their sum is 77/36, which is option (c).

The idea to keep. When a question compares two binomial probabilities, write both out and divide. The binomial coefficients and common powers cancel, and the ratio of p to q is all that is left.

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