Question

A manufacturing machine has a 10% defect rate.If 4 items are chosen at random, what is the probability that at least one will have a defect? Round your answer to three decimal places.

Answer 1

Solution

Step 1

$\begin{gathered} probability\text{ of defective p = 10\% = }\frac{10}{100}\text{ = 0.1} \\ \\ probability\text{ of not defective q = 1 - p = 1 - 0.1 = 0.9} \\ \end{gathered}$

Step2

The probability that at least one will have a defect

= 1 - the probability that noon will have a defect

Step 3

Use the binomial distribution

$\begin{gathered} =1\text{ - }^nC_rp^rq^{n-r} \\ n\text{ = 4, r = 0} \\ \\ =1-^4C_0(0.1)^0(0.9)^{4-0} \\ \\ =\text{ 1 - 1}\times1\times0.9\times0.9\times0.9\times0.9 \\ \\ =\text{ 0.344} \end{gathered}$