Question

Critical Thinking: Interpreting results from a test for smoking It is estimated that roughly half of smokers lie when asked about their smoking involvement. Pulse CO-oximeters may be a way to get information about smoking without relying on patients’ statements. Pulse CO-oximeters use light that shines through a fingernail, and it measures carbon monoxide in blood. These devices are used by firemen and emergency departments to detect carbon monoxide poisoning, but they can also be used to identify smokers. The accompanying table lists results from people aged 18–44 when the pulse CO-oximeter is set to detect a 6% or higher level of carboxyhemoglobin (based on data from “Carbon Monoxide Test Can Be Used to Identify Smoker,” by Patrice Wendling, Internal Medicine News, Vol. 40., No. 1, and Centers for Disease Control and Prevention).

CO-Oximetry Test for Smoking

	PositiveTest Result	Negative Test Result
Smoker	49	57
Non-smoker	24	370

Analyzing the Results

True Positive Based on the results in the table, find the probability that a subject smoke, given that the test result is positive.

Short answer

The probability that a subject is a smoker, given that the test result is positive, is 0.671.

Step-by-step solution

Given information

The table summarizes the test results for smoking.

CO-Oximetry test for smoking

	Positive test result	Negative test result
Smoker	49	57
Non-smoker	24	370

State the formula of conditional probability

Conditional probability of event B occurring, given that event A has already occurred.

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

Find the probabilities

Let event A denotes positive test results and event B denotes that a subject is a smoker.

Total number of subjects are 500.

Using the given information,

$$\begin{gathered} P\left(A\right)=\frac{\text{Number of subjects whose test result is positive}}{\text{Total number of subjects}} \\ =\frac{73}{500} \\ P\left(\text{A and B}\right)=\frac{\text{Number of subjects who are smoker and test result is positive}}{\text{Total number of subjects}} \\ =\frac{49}{500} \end{gathered}$$

Calculate conditional probability 

The true positive is defined as the probability that the subject is a smoker, given the test result is positive.

It is computed as,

$$\begin{gathered} P\left(B\left|A\right.\right)=\frac{P\left(\text{Subject is smoker and test result is positive}\right)}{\text{P}\left(\text{Test result is positive}\right)} \\ P\left(B\left|A\right.\right)=\frac{P\left(A\text{and B}\right)}{P\left(A\right)}...\left(1\right) \end{gathered}$$

Substituting the values in equation (1),

$$\begin{gathered} P\left(B\left|A\right.\right)=\frac{\frac{49}{500}}{\frac{73}{500}} \\ =0.671 \end{gathered}$$

Therefore, the probability that a subject is a smoker, given that the test result is positive, is 0.671.