Question

In Exercises 17–20, refer to the accompanying table showing results from a Chembio test for hepatitis C among HIV-infected patients (based on data from a variety of sources).

	Positive Test Result	Negative Test Result
Hepatitis C	335	10
No Hepatitis C	2	1153

False Positive Find the probability of selecting a subject with a positive test result, given that the subject does not have hepatitis C. Why is this case problematic for test subjects?

Short answer

The probability of selecting a subject with a positive test result, given that the subject does not have Hepatitis C, is equal to 0.00173.

Obtaining a positive result even though the subject does not have Hepatitis C will create unnecessary stress for him/her. He/she may also need to undergo further testing.

Step-by-step solution

Given information

The number of subjects who have Hepatitis C and who do not is tabulated.

The numbers are further divided into positive test results and negative test results.

Conditional probability

The probability of theoccurrence of an event subject to a condition that another event has previously occurred is called theconditional probability of the event.

The following is the expression to compute the conditional probability of B, given A.

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

Compute the probability of the event

Let A be the event of selecting a subject who has Hepatitis C.

Let B be the event of selecting a subject who does not have Hepatitis C.

Let C be the event of selecting a subject who has a positive test result.

Let D be the event of selecting a subject who has a negative test result.

The following table consists of the total frequency under each category:

	Positive Test Result	Negative Test Result	Totals
Hepatitis C	335	10	345
No Hepatitis C	2	1153	1155
Totals	337	1163	1500

The total number of subjects is 1500.

The number of subjects who do not have Hepatitis C is 1155.

The probability of selecting a subject who does not have Hepatitis C is given by:

$P\left(B\right)=\frac{1155}{1500}$

The number of subjects who do not have Hepatitis C and tested positive is 2.

The probability of selecting a subject who tested positive and does not have Hepatitis C is given by:

$P\left(BandC\right)=\frac{2}{1500}$

The probability of selecting a subject who tested positive, given that he/she does not have Hepatitis C, is calculated as follows:

$$\begin{gathered} P\left(C|B\right)=\frac{P\left(BandC\right)}{P\left(B\right)} \\ =\frac{\frac{2}{1500}}{\frac{1155}{1500}} \\ =\frac{2}{1155} \\ =0.00173 \end{gathered}$$

Therefore, the probability of selecting a subject who tested positive, given that he/she did not have Hepatitis C, is equal to 0.00173.

Identifying the problematic case for subjects

The event of testing positive, given that the subject does not have Hepatitis C, will bother the subject by worrying him/her gravely.

The subject might need to undergo further testing, which can lead to unnecessary expenses.