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
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