A faster and simpler diagnostic test for the presence of tuberculosis bacilli, c

Question

A faster and simpler diagnostic test for the presence of tuberculosis bacilli, called “FASTPlaqueTB”, was evaluated in Cape Town, South Africa, and found to have the following reliability. For those subjects known to have the tuberculosis bacilli, the “FASTPlaqueTB” test was positive 70% of the time. For those subjects known not to have the tuberculosis bacilli, the “FASTPlaqueTB” test was negative 99% of the time. Statistics indicate that about 14.2% of the people in Cape Town have the tuberculosis bacilli.

What is the probability that a randomly selected person from Cape Town will have a positive “FASTPlaqueTB” test result?

What is the probability that a person who test positive actually has tuberculosis bacilli?

Explanation / Answer

assume 100,000 were tested (to avid decimals)

14,200 were having TB
70% of 14,200 = 9940 tested positive

85,800 were not having
1% of 85,800 = 858 tested positive

a. P(positive test result) = (9940+858)/100,000 = 10.8% <-----

b. P(has TB | tested positive) = 9940/(9940+858) = 92.1%

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