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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锟糳enotes 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,

$P\left(A鈥�鈥�\text{and}鈥�鈥�B\right)=鈥�\text{Probability}鈥�\text{of}鈥�鈥�\text{A}鈥�鈥�\text{and}鈥�鈥�\text{B}鈥�鈥�\text{both}鈥�鈥�\text{occurring}$

$P\left(A\right)=\text{Probability}鈥�鈥�\text{of}鈥�鈥�\text{occurring}鈥�鈥�\text{event}鈥�鈥�A$

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 summarizedin the following table,

The probability that a subject does not smoke, given that the test yields a negative result. Here,define the two events as:A be the event that the test yields a negative result, and B be the event that the subject does not smoke.

The probabilitythat a subject does not smoke, given that the test yields a negative result,$P\left(B|A\right)$.

By using conditional probability,

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

The probability that the test yields a negative result is,

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{subjects with negative}鈥�鈥�\text{test}鈥�鈥�\text{reults}}{\text{Total subjects surveyed}} \\ =\frac{427}{500} \end{gathered}$$

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

$$\begin{gathered} P\left(A鈥�鈥�\text{and}鈥�鈥�B\right)=\frac{\text{Number of subjects who does not}\text{smoke and yield negative test result}}{\text{Total}} \\ =\frac{370}{500} \end{gathered}$$

The negative predictive value is the probability that a subject does not smoke, given that the test yields a negative result.

Substitute $P\left(\text{subject does not smoke and yield neagtive test result}\right)$and$P\left(\text{Negative result of test}\right)$ in equation (1),

$$\begin{gathered} P\left(\text{B|A}\right)=\frac{P\left(A鈥�鈥�鈥�\text{and}鈥�鈥�鈥�B\right)}{P\left(A\right)} \\ =\frac{\frac{370}{500}}{\frac{427}{500}} \\ =\frac{370}{427} \\ =0.8665 \end{gathered}$$

Thus, the probability that a subject smoke, given that the test yields a negative result is 0.866.