Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If μ and σ² represent mean and variance of X, respectively, then 10(μ²+σ²) is equal to :

Solution

Maths sir · 4 steps

Mean squared plus variance is just E(X²), so find that one number and you are done.

1. Spot the shortcut. Variance is σ² = E(X²) - μ², so μ² + σ² = E(X²). You only need E(X²).
2. Build the distribution. Drawing 4 apples one by one without replacement gives the same probabilities as choosing 4 of the 10 apples at once. There are C(10, 4) = 210 such choices, and X can be 0, 1, 2 or 3.

   P(X = r) = C(3, r)C(7, 4-r)/C(10, 4)

   X	P(X)	X · P(X)	X² · P(X)
   0	(1 × 35)/210 = 35/210	0	0
   1	(3 × 35)/210 = 105/210	105/210	105/210
   2	(3 × 21)/210 = 63/210	126/210	252/210
   3	(1 × 7)/210 = 7/210	21/210	63/210

3. Add the columns. The probabilities add to 210/210 = 1, a good check. Then μ = 252/210 = 1.2 and E(X²) = 420/210 = 2, so σ² = 2 - 1.44 = 0.56.
4. So. μ² + σ² = 1.44 + 0.56 = 2, and 10(μ² + σ²) = 10 × 2 = 20, option (a).

The idea to keep. Whenever a question asks for μ² + σ², read it as E(X²). And drawing without replacement means counting with combinations, not a binomial with a fixed p.

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