Question

A shipment of 10 printers contains 4 that are defective. Find the probability that a sample of size 4, drawn from the 10, will not contain a defective printer. The probability is (Type an integer or a simplified fraction.)

Answer 1

Let X be the random variable that denotes the number of defective printers in the sample.

A shipment of 10 printers contains 4 that are defective. Therefore, N = 10 and M = 4.

A sample of size 4 is drawn. Therefore, n = 4.

X $\sim$ Hypergeometric (N = 10, M = 4, n = 4)

The pmf of X is

P(X = x) = (^MC_x * ^{N - M}C_{n - x}) / ^NC_n ; x = 0, 1, ......, min(M, n).

              = 0                                     ; otherwise

So, the pmf of X is

P(X = x) = (⁴C_x * ⁶C_{4 - x}) / ¹⁰C₄ ; x = 0, 1, 2, 3, 4.

              = 0                                ; otherwise

P(X = 0) = (⁴C₀ * ⁶C_{4 - 0}) / ¹⁰C₄

                 = 1 / 14

Therefore, the probability that the sample will not contain a defective printer is 1/14.