The ELISA test
In 1% of cases, an HIV test gives a positive result for someone who is HIV negative. How likely is it that someone who tests positive has HIV?
Problem
This resource is part of the collection Probability and Evidence.
This problem considers an alternative way in which understanding the evidence is important, here considering medical tests.
Alternatively, click below to read a description.
The correct use of probability and statistics is fundamental to various applications, one of which is medical testing.
The ELISA test, an indicator for HIV, is a good example of this. There is a 1% chance that if you are HIV negative, you will get a positive result. However, this does not mean that if you get a positive result, then you have a 1% chance of being HIV negative.
The ELISA (Enzyme Linked Immunosorbent Assay) tests can be used to detect whether someone is HIV positive. These tests are cheap and easy to administer, but they are not always accurate.
In particular, for someone without HIV, there is a 1% chance that the test will record a positive result, called a false-positive.
Why is this not the same as saying "a positive result means there is a 99% chance of being infected"?
In low-risk groups, the rate of infection is approximately 1 in 10,000.
Virtually all people with HIV record a positive result: the probability of a false-negative result is negligible.
How could you use this new information to calculate the probabilty that someone who gets a positive result has HIV?
Are there any tables or diagrams that might help you represent the information?
Can you consider what you might expect to happen to 10,000 random people?
When you have thought about these questions, click below for some suggestions:
You could draw a two-way (contingency) table like this:
| Positive Test Result | Negative Test Result | Total | |
|---|---|---|---|
| Person is HIV Positive | |||
| Person is HIV Negative | |||
| Total | 10 000 |
Can you use this table to work out the probability that someone who tests positive actually has HIV?
Does this result surprise you?
Why is this test useful, despite the number of false-positives it produces?
Student Solutions
This was a tricky problem but Zach sent in a very comprehensive answer. He has used a table to show the probabilities and has looked at what would happen if the rate of infection was higher. You can see his full solution here.
Teachers' Resources
The classroom activities The Dog Ate My Homework and Who is Cheating? contain similar ideas.