What Was the Accuracy of HIV Testing for Marriage Licenses in Illinois in 1988?

[Darth Frodo]

Homework Statement

In 1988, the state of Illinois required HIV testing for a couple to obtain a marriage
license. The HIV testing at the time consisted of two separate tests, the ELISA
test and the Western Blot test. The Elisa test was signicantly less expensive. A
person who is HIV positive would test positive under the ELISA test 95% of the
time. A person who is HIV negative would test positive under the ELISA test 99%
of the time. In 1988, it was estimated that the percentage of people applying
for a marriage license that were actually HIV positive was 1%.

If someone test positive on the ELISHA test, then that person is given the West-
ern Blot test. A person who is HIV positive will test positive on the Western
Blot test 99% of the time, while a person who is HIV negative will test positive
on the Western Blot test 5% of the time. What is the conditional probability that someone is HIV positive given that the person tests positive on both the Western Blot and the ELISHA test?

The Attempt at a Solution

Basically I drew a tree diagram to figure this out, but it seems very low. I would appreciate it if someone could tell me where I'm going wrong.

If anyone has any questions on notation, please ask

 

[mfb]

A person who is HIV negative would test positive under the ELISA test 99%

I think this should be "negative" (or 1% instead of 99%), otherwise the test would be very strange.
While this should change your result to a larger value, small probabilities to have some rare disease are not uncommon in medical tests in general.

 

[Darth Frodo]

Yes, I think you're right. I completely misread the question. Although, I think my confusion is rather valid.