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-oximeters 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

	Positive Test Results	Negative Test Results
Smoker	49	57
Non-smoker	24	370

Confusion of the Inverse Find the following values, then compare them. In this case, what is confusion of the inverse?

$P\left(Smoker|Positive test result\right)$

$P\left(Positive test result| smoker\right)$

Short answer

The probabilities are:

$P\left(\text{Smoker}\left|\text{Positive} \text{test} \text{result}\right.\right)=0.671$

$P\left(\left.\text{Positive} \text{test} \text{result}\right|\text{Smoker}\right)=0.462$

The confusion of the inverse is:

$P\left(\text{Smoker}\left|\text{Positive} \text{test} \text{result}\right.\right)\ne P\left(\left.\text{Positive} \text{test} \text{result}\right|\text{Smoker}\right)$

Step-by-step solution

Given information

The results from people aged 18-44 to detect the level of carboxyhemoglobin (based on the data from the Carbon Monoxide Test which was used to identify smokers).

Describe conditional probability

The conditional probability of an event is the probability obtained with the additional information that some other event has already occurred.

Notation for conditional probability$P\left(B|A\right)$denotes the conditional probability of event B occurring, given that event A has already occurred.

Formula for conditional probability is as,

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

Summarize the given information

The test for smoking involves 500 people. Out of the 500 people, 106 are smokers and 394 are non-smokers. And, 73 people get the positive test results and 427 get the negative test results.

The given information is summarized as follow in the table.

	Positive test results	Negative test results	Total
Smoker	49	57	106
Non-smoker	24	370	394
Total	73	427	500

Define the two events

The expression $P\left(\text{Smoker| Positive} \text{test} \text{result}\right)$means:the probability that the subject is a smoker, given that the test yields a positive result.

For simplicity,define the two events as,

A be the event that the test yields positive results and B be the event that the subject is a smoker.

Find the probability that the subject is a smoker, given that the test yields positive results.

The probabilitythat the subject is a smoker, given that the test yields a positive result is $P\left(B|A\right)$.

The probability that the test yields a positive result, given that the subject is a smoker, is $P\left(A|B\right)$.

By using conditional probability,

$$\begin{gathered} P\left(B\left|A\right.\right)=\frac{P\left(A \text{and} B\right)}{P\left(A\right)} ...\left(1\right) \\ P\left(A\left|B\right.\right)=\frac{P\left(A \text{and} B\right)}{P\left(B\right)} ...\left(1\text{'}\right) \end{gathered}$$

Find the probability that the test yields positive results

The probability that the test yields a positive result is,

$$\begin{gathered} P\left(A\right)=\frac{\text{Subjects with positive} \text{test} \text{reults}}{\text{Total subjects}} \\ =\frac{73}{500}...\left(2\right) \\ P\left(B\right)=\frac{\text{Subjects}\text{who}\text{smoke}}{\text{Total}\text{subjects}} \\ =\frac{106}{500}...\left(3\right) \end{gathered}$$

Find the probability that the subject is a smoker and the test yields a positive test result

The probability that the subject is a smoker and the test yields a positive test result is,

$$\begin{gathered} P\left(A \text{and} B\right)=\frac{\text{Subjects who smokes and yield positive test result}}{\text{Total subjects}} \\ =\frac{49}{500}...\left(4\right) \end{gathered}$$

Find the probability PSmoker|Positive  test  result

Substitute the probability values found in equations (2) and (4) in equation (1).

$$\begin{gathered} P\left(\text{A|B}\right)=\frac{P\left(A \text{and} B\right)}{P\left(B\right)} \\ =\frac{\frac{49}{500}}{\frac{106}{500}} \\ =\frac{49}{106} \\ =0.4623 \end{gathered}$$

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

Interpret the confusion of the inverse

By observing these two cases,the confusion of the inverse is as,

$P\left(\text{Smoker| Positive} \text{test} \text{result}\right)\ne P\left(\text{Positive} \text{test} \text{result|smoker}\right)$

Here, the confusion of the inverse states that “the probability that the subject is a smoker, given that the test yields a positive result” is not equal to “the probability of event that the subject tests positive, given that he is a smoker”.