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 Negative Find the probability of selecting a subject with a negative test result, given that the subject has hepatitis C. What would be an unfavorable consequence of this error?

Short answer

The probability of selecting a subject with a negative test result, given that the subject does has Hepatitis C, is equal to 0.0290.

Obtaining a negative result even though the subject has Hepatitis C will delay the necessary treatment required for the subject. It may lead to the deterioration in his/her health.

Step-by-step solution

Given information

The frequencies of subjects who have Hepatitis C and who do not are tabulated.

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

Conditional probability

The conditional probability of an event describes the likelihood of the event with reference to some prior condition.

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

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

Calculation

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 equal to 1500.

The number of subjects who have Hepatitis C is 345.

The probability of selecting a subject who has Hepatitis C is given by:

$P\left(A\right)=\frac{345}{1500}$

The number of subjects who have Hepatitis C and tested negative is equal to 10.

The probability of selecting a subject who tested negative and has Hepatitis C is given by:

$P\left(AandD\right)=\frac{10}{1500}$

The probability of selecting a subject who tested negative, given that he/she has Hepatitis C, is calculated as follows:

$$\begin{gathered} P\left(D|A\right)=\frac{P\left(AandD\right)}{P\left(A\right)} \\ =\frac{\frac{10}{1500}}{\frac{345}{1500}} \\ =\frac{10}{345} \\ =0.0290 \end{gathered}$$

Therefore, the probability of selecting a subject who tested negative, given that he/she has Hepatitis C, is equal to 0.0290.

Identifying the consequences for the error

The event of testing negative, given that the subject has Hepatitis C, will delay the onset of the treatment of the disease and can worsen the health of the subject.