Question

A certain virus infects one in every 300 300 people. A test used to detect the virus in a person is positive 90 90​% of the time when the person has the virus and 10 10​% of the time when the person does not have the virus.​ (This 10 10​% result is called a false positive​.) Let A be the event​ "the person is​ infected" and B be the event​ "the person tests​ positive." ​(a) Using​ Bayes' Theorem, when a person tests​ positive, determine the probability that the person is infected. ​(b) Using​ Bayes' Theorem, when a person tests​ negative, determine the probability that the person is not infected. (ROUND TO FOUR DECIMALS)

Answer 1

P(Infected) = 1/300

P(Not infected) = 299/300

P(positive test) when infected = 0.90

P(negative test) when infected = 0.10

P(positive test) when not infected = 0.10

P(negative test) when not infected = 0.90

a) P(positive and infected) = 0.90 * 1/300 = 0.003

P(positive test) = 1/300 * 0.90 + 0.10 * 299/300 = 0.10267

Probability that the person is infected given that he tests positive, P(Infected | positive test) = P(positive and infected) / P(positive test)

= 0.003 / 0.10267

= 0.02922

b) P(Negative and not infected) = 0.90 * 299/300 = 0.897

P(Negative test) = 0.90 * 299/300 + 0.10 * 1/300 = 0.89733

Probability that the person is not infected given that the person tests​ negative, P( doesn’t have virus | negative test) = P(Negative and not infected) / P(Negative test)

= 0.897 / 0.89733

= 0.9996

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