From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is σ², then 96 σ² is equal to __________.

Solution

Maths sir · 3 steps

Write out the full distribution of X, then variance is the mean of X² minus the square of the mean.

1. Set up X. There are 3 defective and 7 good items, and 5 are drawn without replacement, so X can be 0, 1, 2 or 3. The number of ways to choose any 5 items is C(10, 5) = 252.

   X	Ways to choose	P(X)
   0	C(7, 5) = 21	1/12
   1	C(3, 1)C(7, 4) = 105	5/12
   2	C(3, 2)C(7, 3) = 105	5/12
   3	C(3, 3)C(7, 2) = 21	1/12

2. Two expectations. E(X) = (0 + 5 + 10 + 3)/12 = 3/2. E(X²) = (0 + 5 + 20 + 9)/12 = 17/6.

   σ² = E(X²) - (E(X))² = 17/6 - 9/4 = 7/12

3. So. 96σ² = 96 × 7/12 = 56.

The common slip: 100.8. Treating the draws as binomial with p = 0.3 gives σ² = 5 × 0.3 × 0.7 = 1.05 and 100.8, not 56. The items are not replaced, so the trials are not independent.

The idea to keep. Drawing without replacement is not binomial. Either list the distribution, or use the binomial variance multiplied by the correction (N-n)/(N-1).

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