# Probability of having a disease given a series of test results

A person completes medical test 4 times, resulting in (first=negative, second=positive, third=negative and fourth=negative).

What is the probability that that the person in fact has the Disease after having done the four tests?

Prevalance  = 15% ... which I believe is P(Disease)

The first three tests (negative, positive, negative) have

Sensitivity = 60%  ... which I believe  is  P(Positive|Disease)
Specificity = 99.5% ... which  I believe is P(Negative|No Disease)

the final test (negative) has Sensitivity that is increased to 80%, thus:

Sensitivity = 80%  ... which I believe  is  P(Positive|Disease)
Specificity = 99.5% ... which  I believe is P(Negative|No Disease)

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Kav10
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• Hey Kav10, thank you! A quick question just quickly looking at this, is it ~40% or 4%? I think ~40% correct? 0.40.. I'll look in detail a later tonight.

• Yes, you are correct, it is %40.7. The calculation and the result should be correct. The percentage mentioned in the text is a typo. Sounds good :-)

• Thanks. Everything makes sense to me and the answer looks great. Appreciate it! I did have a question if you're willing to answer: what would be an example of a real life scenario where these tests wouldn't be independent? The best I could come up with is for instance if the same input specimen was used (and let's say potentially contaminated) then other tests would be dependent on this contamination. Anything else readily come to your mind?

• p.s. would you mind updating the % to 40% in case i come back to reference this at some point or someone else decides to get it . Looks like I can't update it.

• Sure. An example for a real-life scenario where these tests wouldn't be independent is when a patient undergoes multiple tests to diagnose a specific disease, and the results of one test affect the probability of a positive result on the subsequent test. For example, if a patient has a positive result on a preliminary screening test for a certain disease, they may be more likely to have a positive result on a subsequent confirmatory test.

• In this scenario, the results of the screening test would affect the probability of a positive result on the confirmatory test, so the two tests would not be independent.

• And to your point, yes, you are correct. Using the same input specimen for multiple tests, such as a blood sample, and if that specimen becomes contaminated, it can affect the results of all the subsequent tests. This would make the results of those tests dependent on each other, as the contamination of the specimen would affect the probability of a positive or negative result on all the tests.

• Another example would be the use of a diagnostic imaging test, such as an X-ray or MRI, which can provide information that influences the results of subsequent tests. For example, if a chest X-ray shows a mass, it could increase the likelihood of a positive result on a subsequent biopsy or blood test for cancer markers.

• Another example could be if a person has a family history of the disease, or if they have certain risk factors that would increase the likelihood of them having the disease. In this case, the results of one test may influence the results of subsequent tests. For example, if a person has a family history of a disease, they may be more likely to test positive for that disease even if they don't actually have it, which could affect the results of subsequent tests.

• Another example could be if the tests are measuring different aspects of the same underlying condition. For example, if a person is tested for both high blood pressure and diabetes, the results of the first test may influence the results of the second test. If a person tests positive for high blood pressure, they may be more likely to test positive for diabetes as well, even if they don't actually have it.

• However, in your question, assuming those are let's say COVID test, they are independent, if the same test is used.