A very large batch of components has arrived at a distributor. The batch can be characterized as acceptable only if the proportion of defective components is at the most .06. The distributor decides to randomly select 7 components and to accept the batch only if the number of defective components in the sample is no more than 1 item found defective. what is the probability of rejecting the batch?

Solution

Let $X$ denote the number of defectives within 7 selected components.

Let $p=0.06$ be the proportion of defectives. Then the random variable $X$ follows a binomial distribution. That is $X\sim B(7,0.06)$.

The probability mass function of $X$ is
$$ \begin{aligned} P(X=x) &= \binom{7}{x} (0.06)^x (1-0.06)^{7-x},\\ & \qquad \; x=0,1,\cdots, 7. \end{aligned} $$

We accept the batch only if no more than 1 item found defective. That is $X\leq 1$.

Thus the probability of rejecting the batch is

$$ \begin{aligned} P(X > 1) &= 1- P(X\leq 1)\\ &= 1- \bigg(P(X=0) + P(X=1)\bigg)\\ &=1-\bigg(\binom{7}{0} (0.06)^{0} (1-0.06)^{7-0}+\binom{7}{1} (0.06)^{1} (1-0.06)^{7-1}\bigg)\\ & = 1-0.9382\\ &= 0.0618 \end{aligned} $$