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The results from people aged 18-44 to detect the level of carboxyhemoglobin are based on the data from 鈥淐arbon Monoxide Test鈥� used to identify smokers.

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

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

Formula for conditional probability is as,

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

Where,

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

The above information is summarized in the following table,

Positive predictive value is the probability that a subject smokes,given the test yields a positive result.

Define the two events as,

A be the event that the test yields a positive result, and B be the event that the subject smokes.

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

By using conditional probability,

$P\left(\text{B|A}\right)=\frac{P\left(A鈥�鈥�鈥�\text{and}鈥�鈥�鈥�B\right)}{P\left(A\right)}鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�...\left(1\right)$

The probability that the test yields a positive result is,

$$\begin{gathered} P\left(A\right)=\frac{\text{Number of positive}鈥�鈥�\text{test}鈥�鈥�\text{results}}{\text{Total}\text{number of surveyed subjects}} \\ =\frac{73}{500} \end{gathered}$$

The probability that a subject does not smoke and had negative test result is,

$$\begin{gathered} P\left(A鈥�鈥�\text{and}鈥�鈥�B\right)=\frac{\text{Number of subject who smoke and had positive test result}}{\text{Total number of subjects surveyed}} \\ =\frac{49}{500} \end{gathered}$$

Positive predictive value is the probability that a subject smokes, given that the test yields a positive result.

Substitute the probability of $P\left(\text{subject smoke and yield positive test result}\right)$and$P\left(\text{Positive result of test}\right)$ in equation (1),

$$\begin{gathered} P\left(B|A\right)=\frac{P\left(A鈥�鈥�鈥�\text{and}鈥�鈥�鈥�B\right)}{P\left(A\right)} \\ =\frac{\frac{49}{500}}{\frac{73}{500}} \\ =\frac{49}{73} \\ =0.6712 \end{gathered}$$

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