The standard test given to determine if you have tuberculosis is called a TB tes

Question

The standard test given to determine if you have tuberculosis is called a TB test. If you test positive for tuberculosis, you will have a reaction within 48 hours. According to the Centers for Disease Control and Prevention, 0.005% of the population in the United States have tuberculosis and the test is 99% effective for determining whether a person does not have tuberculosis.

For this problem, we will assume that the TB test has a 0.1% false-negative result. In other words, 0.1% of the population with tuberculosis have a negative test result.

1.) What is the probaility that an individual selected at random has tuberculosis given that the test was positive?

2.) What is the probability that an individual selected at random has a positive test result, given that person has tuberculosis?

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Explanation / Answer

Ans:

P(TB)=0.00005

P(not TB)=1-0.00005=0.99995

P(positive/TB)=0.99

P(negative/TB)=1-0.99=0.01

As,false negative =0.1%,so

P(positive/not TB)=0.001

P(negative/not TB)=1-0.001=0.999

1)

P(positive)=0.99*0.00005+0.001*0.99995=0.00105

P(TB/positive)=P(positive/TB)*P(TB)/P(positive)=0.99*0.00005/0.00105=0.0471

2)

P(positive/TB)=0.99

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