Question

A manufacturing company performs a quality control analysis on the ceramic tile it produces. Suppose a batch of 22 tiles has 6 defective tiles. If 5 tiles are sampled at random, what isthe probability that exactly 1 of the sampled tiles is defective?How many ways can 5 tiles be selected from 22 tiles?ways(Type a whole number)Help Me Solve ThisView an ExampleGet More HelpClear AllCheck Answer

Answer 1

ANSWER:

The probability is 0.4147

26334 ways

STEP-BY-STEP EXPLANATION:

The first thing is to calculate numbers of ways of choosing 5 out of 22 (without replacement), just like this:

$\begin{gathered} \text{nCr}=\frac{n!}{r!\cdot(n-r)!} \\ \text{ replacing} \\ 22C5=\frac{22!}{5!\cdot(22-5)!}=26334 \end{gathered}$

Out of 22 tiles, 6 are defective and remaining 16 are not defective.

Let a = number of defective among randomly selected 5 tiles.

Here a follows hypergeometric distribution with parameters N = 22, k = 6, n = 5.

PMF is given: P(a = k), in this case:

$P(a=1)=\frac{(\text{choose 1 of 6 defective)}\cdot(\text{choose 4 of 16 not defective) }}{(\text{select 5 from 22 total)}}$

replacing:

$P(a=1)=\frac{\frac{6!}{1!\cdot(6-1)!}\cdot\frac{16!}{4!\cdot(16-4)!}}{\frac{22!}{5!\cdot(22-5)!}}=\frac{6\cdot1820}{26334}=0.4147$