Question

In Exercises 21–24, use these results from the “1-Panel-THC” test for marijuana use, which is provided by the company Drug Test Success: Among 143 subjects with positive test results, there are 24 false positive results; among 157 negative results, there are 3 false negative results. (Hint: Construct a table similar to Table 4-1, which is included with the Chapter Problem.)

Testing for Marijuana Use If one of the test subjects is randomly selected, find the probability that the subject tested negative or used marijuana.

Short answer

The probability of selecting a subject who tested negative or used marijuana is equal to 0.92.

Step-by-step solution

Given information

The given data shows the number of positive and negative drug test results of the subjects.

The values are categorized as true and false results.

Addition rule of probability

For two events A and B, the probability of occurrence of only A, only B, or both is computed as shown below:

$P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}B\right)$

Here, $P\left(A\text{and}B\right)$ is the probability of A andB occurring together.

Calculation

The table below shows the number of subjects that fall into each category:

	True Result	False Result	Total
Subject Tested Positive	143 – 24 = 119	24	143
Subject Tested Negative	157 – 3 =154	3	157
Total	273	27	Grand Total=300

The total number of subjects is equal to 300.

Let E be the event of selecting a subject who tested negative.

Let Fbe the event of selecting a subject who used marijuana.

The number of subjects who tested negative is equal to 157.

The number of subjects who used marijuana is equal to:

$$\begin{gathered} \text{True}\text{Positive}+\text{False}\text{Negative}=119+3 \\ =122 \end{gathered}$$

So, the corresponding probabilities are:

$$\begin{gathered} P\left(E\right)=\frac{157}{300} \\ P\left(F\right)=\frac{122}{300} \end{gathered}$$

The number of subjects who tested negative and used marijuana is equal to the number of subjects who tested false. Thus, it is equal to 3.

$P\left(E\text{and}F\right)=\frac{3}{300}$

Theprobability of selecting a subject who tested negative or used marijuana is equal to:

$$\begin{gathered} P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right) \\ =\frac{157}{300}+\frac{122}{300}-\frac{3}{200} \\ =\frac{276}{300} \\ =0.92 \end{gathered}$$

Therefore, the probability of selecting a subject who tested negative or used marijuana is equal to 0.92.