Question

Reminder: You obtain a positive test result for HIV. There is no reason to believe that you should have a higher prior probability of being HIV positive than the average the average person in Australia. In Australia, about 30,000 people out of 24 million people are HIV positive. The test has a false negative rate of 0.2% (i.e., the probability of obtaining a negative result for a person who is HIV positive is 0.002) and a false positive rate of 2.5% (i.e., the probability of obtaining a positive result for a person who is HIV negative is 0.025). After obtaining this test result, what are the posterior odds in favour of you being HIV positive?

A. 0.001 (this corresponds to odds of about 1 to 911 that you are HIV positive)
B. 0.015 (this corresponds to odds of about 1 to 67 that you are HIV positive)
C. 0.063 (this corresponds to odds of about 1 to 16 that you are HIV positive)
D. 0.072 (this corresponds to odds of about 1 to 14 that you are HIV positive)
E. 0.050 (this corresponds to odds of about 1 to 20 that you are HIV positive)

Answer 1

Let HIV be the event that a random person is HIV positive. Let N and P be the event that the person tested negative and positive respectively.

Probability that a random person is HIV positive, P(HIV) = 30000 / 24 x 10⁶ = 0.00125

Probability that a random person is not HIV positive (HIV negative), P(~HIV) = 1 - 0.00125 = 0.99875

Probability of obtaining a negative result for a person who is HIV positive = P(N | HIV) = 0.002

Probability of obtaining a positive result for a person who is HIV negative = P(P | ~HIV) = 0.025

Probability of obtaining a positive result for a person who is HIV positive = P(P | HIV) = 1 - P(N | HIV) = 1 - 0.002 = 0.998

Probability of obtaining a positive result = P(P)

= P(HIV) P(P | HIV) + P(~HIV) P(P | ~HIV) (By law of total probability)

= 0.00125 * 0.998 + 0.99875 * 0.025

= 0.02621625

Probability of being HIV positive given positive results = P(HIV | P)

= P(P | HIV) * P(HIV) / P(P) (By Bayes theorem)

=  0.998 * 0.00125 / 0.02621625

= 0.04758499

Posterior odds in favour of you being HIV positive = 0.04758499 / (1 - 0.04758499) = 0.05

E. 0.050 (this corresponds to odds of about 1 to 20 that you are HIV positive)