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 positive or did not use marijuana.

Short answer

The probability of selecting a subject who tested positive or did not use marijuana is equal to 0.99.

Step-by-step solution

Given information

The number of positive and negative drug test results of the subjects is provided.

The frequencies 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, the probability of occurrence of both A and B is given by $P\left(A\text{and}B\right)$.

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 positive.

Let F be the event of selecting a subject who did not use marijuana.

The number of subjects who tested positive is equal to 143.

The number of subjects who did not use marijuana is equal to:

$$\begin{gathered} \text{False}\text{Positive}+\text{True}\text{Negative}=24+154 \\ =178 \end{gathered}$$

So, the corresponding probabilities are:

$$\begin{gathered} P\left(E\right)=\frac{143}{300} \\ P\left(F\right)=\frac{178}{300} \end{gathered}$$

The number of subjects who tested positive but did not use marijuana is equal to the number of subjects who tested false positive. Thus, it is equal to 24.

$P\left(E\text{and}F\right)=\frac{24}{300}$

The probability of selecting a subject who tested positive or did not use marijuana is equal to:

$$\begin{gathered} P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right) \\ =\frac{143}{300}+\frac{178}{300}-\frac{24}{300} \\ =\frac{297}{300} \\ =0.99 \end{gathered}$$

Therefore, the probability of selecting a subject who tested positive or did not use marijuana is equal to 0.99.