College Algebra, Chapter 10, 10.4, Section 10.4, Problem 26

An assembly line that manufactures fuses for automotive use is checked every hour to ensure the quality of the finished product. Ten fuses are selected randomly, and if anyone of the ten is found to be defective, the process is halted and the machines are re-calibrated. Suppose that at a certain time $5\%$ of the fuses being produced are actually defective. What is the probability that the assembly line is halted at that hour's quality check?

Recall that the formula for the binomial probability is given by

$C(n,r) p^r q^{n-r}$

In this case, the probability of success is $p =0.05$ and the probability of failure is $q = 1-p = 0.95$.

To solve this more easily, we will apply the complement to the probability that none of the ten fuses selected are not defective. Thus, the probability that at least one of ten fuses are defective is


$
\begin{equation}
\begin{aligned}

=& 1 - \left[C(10,0) (0.05)^0 (0.95)^{10-0}\right]
\\
\\
=& 1 - [0.5987]
\\
\\
=& 0.4013

\end{aligned}
\end{equation}
$