Question

Silk scarves are produced in a factory. Quality control investigations find that 0.5% of the scarves produced have flaws, which reduce their value. A sample of 30 scarves is selected from the factory. Calculate the probability that the sample has:Exactly one flawed scarfNo flawed scarvesMore than three flawed scarves

Answer 1

The binomial probability is determined by the formula

$\begin{gathered} P_x=\binom{n}{x}p^xq^{n-x} \\ \text{where} \\ P\text{ is the binomial probability} \\ x\text{ is the number of times for a specific outcome within }n\text{ trials} \\ p\text{ is the probability of success on a single trial} \\ q\text{ is the probability of failure on a single trial} \\ n\text{ is the number of trials} \end{gathered}$

Here, we will assume that finding the flaw is the "success" of the trial hence the following given

$\begin{gathered} p=0.05 \\ q=0.95 \\ n=30 \end{gathered}$

Finding the probability of one flawed scarf

$\begin{gathered} P(X=1)=\binom{30}{1}(0.05)^1(0.95)^{30-1} \\ P(X=1)=0.3389 \end{gathered}$

Finding the probability of no flawed scarves

$\begin{gathered} P(X=0)=\binom{30}{0}(0.05)^0(0.95)^{30-0} \\ P(X=0)=0.2146 \end{gathered}$

Finding the probability of more then three flawed scarves

$\begin{gathered} P(X>3)=1-P(X\le3) \\ P(X>3)=1-0.93922843869 \\ P(X>3)=0.0608 \end{gathered}$