Associated Topics || Dr. Math Home || Search Dr. Math

Test for Tuberculosis

```
Date: 07/28/97 at 14:21:33
From: Rachel Strickland
Subject: Probability

A test for tuberculosis was given to 1000 subjects, 8 percent of whom
were known to have tuberculosis. For the subjects who had
tuberculosis, the test indicated tuberculosis in 90 percent of the
subjects,was inconclusive for 7 percent and negative for 3 percent.
For the subjects who did not have tuberculosis, the test indicated
tuberculosis in 5 percent, was inconclusive for 10 percent, and
was negative for the remaining 85 percent.

What is the probability of a randomly selected person having
tuberculosis given that the test indicates tuberculosis?
Of not having tuberculosis given that the test was inconclusive?
```

```
Date: 07/28/97 at 17:11:57
From: Doctor Anthony
Subject: Re: Probability

This is a Bayesian probability problem, in that we are given that the
test result is known. i.e. positive test result in first instance.

For this situation, we draw up a table as shown:

Have T.B.(8%)       Do not have T.B.(92%)
----------------------------------------------
.08 x .90               .92 x .05              Test is positive
----------------------------------------------
.08 x .07               .92 x .10              Test inconclusive
----------------------------------------------
.08 x .03               .92 x .85              Test negative
----------------------------------------------

Now we can read off probabilities knowing that the sample space is
restricted to just one line of the table.

(1) Probability that person has T.B. given that test is positive

.08 x .90                 .072           36
=   ---------------------  =  -------------  =  ---- =  0.6102
.08 x .90 + .92 x .05     .072 + .046        59

(2) Probability of having T.B. given that test is inconclusive

.08 x .07                  .0056           7
=  --------------------   =  -------------- =  ----- =  0.057377
.08 x .07 + .92 x .10      .0056 + .092       122

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search