A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If σ² is the variance of X, then 100 σ² is equal to ________.

Solution

Maths sir · 3 steps

Write out the distribution of X, then use mean of the squares minus square of the mean.

1. Count the ways. Three balls from 10 can be chosen in C(10, 3) = 120 ways. For X = k white balls, choose k from the 4 white and 3-k from the 6 black.

   X	Ways	P(X)
   0	C(4, 0)C(6, 3) = 20	20/120
   1	C(4, 1)C(6, 2) = 60	60/120
   2	C(4, 2)C(6, 1) = 36	36/120
   3	C(4, 3)C(6, 0) = 4	4/120

2. Mean and mean square. E(X) = (0 + 60 + 72 + 12)/120 = 144/120 = 1.2. E(X²) = (0 + 60 + 144 + 36)/120 = 240/120 = 2.

   σ² = E(X²) - (E(X))² = 2 - 1.44 = 0.56

3. So. 100σ² = 100 × 0.56 = 56.

The common slip: 72. Treating the draws as binomial with p = 0.4 gives σ² = 3 × 0.4 × 0.6 = 0.72, so 72. That formula needs replacement, and here the balls are not put back.

The idea to keep. Balls drawn together means no replacement, so X is not binomial. Build the table from combinations and use σ² = E(X²) - (E(X))².

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