Chapter 10, Problem 12RE

Solution

To explain : The two kinds of contamination are independent or not.

The two kinds of contamination are not independent.

Given information :

In 2007 Consumer Reports purchased samples of chicken offered for sale in $23$ states and checked them for bacterial contamination. The magazine found that $81\%$ of the packages were contaminated with campylobacter, $15\%$ with salmonella, and $13\%$ with both.

Formula used :

Two events $A$ and $B$ are independent if $P\left(\left.B\right|A\right)=P\left(B\right)$ .

  $P\left(\left.B\right|A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}$

Calculation :

Let $A$ be the event that the packages were contaminated with campylobacter and $B$ be the event that the packages were contaminated with salmonella.

According to the given information,

  $\begin{array}{c}P\left(A\right)=0.81\\ P\left(B\right)=0.15\\ P\left(A\cap B\right)=0.13\end{array}$

So,

  $\begin{array}{c}P\left(\left.B\right|A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}\\ =\frac{0.13}{0.81}\\ \approx 0.16\\ \ne P\left(B\right)\end{array}$

Therefore, the two kinds of contamination are not independent.