4 ms·

I'm not sure you phrased the problem correctly. If we follow your explanation, then the probability of having the disease is indeed 99%. If you want to show th

I'm not sure you phrased the problem correctly. If we follow your explanation, then the probability of having the disease is indeed 99%. If you want to show the implication of Bayes' Theorem then you need to be more precise : Say you have a 1% of false positive and false negative rates (99% reliability) and 1% of the population is sick. If you are tested positive, then the probability of being sick is much less than 99%.

sykh 8y ago
I updated the problem. Sorry for the mistake.
thaumasiotes 8y ago
> If we follow your explanation, then the probability of having the disease is indeed 99%. This is not correct; the probability of having the disease is unknown. He didn't say what he meant by the test being "99% accurate", but that doesn't mean you can just make your own assumption. Note that in your more precisely specified scenario, when the test has 99% reliability, it is perfectly true that "99% of the time the test gives a correct result", which immediately disproves the claim that, if we follow that definition, the probability of having the disease given a positive test result is 99%.
- cdancette 8y ago
  The problem is that "99% the time gives a correct result" is imprecise. It can be understood as both: - p(sick|positive) = 0.99 - p(positive|sick) = 0.99 We get totally different results, the first one is obvious (99% change of being sick), and the second one needs Bayes' Theorem (and is the one we want to use).
  - thaumasiotes 8y ago
    I would only interpret "the test gives a correct result 99% of the time" to mean that out of every 100 test results, 99 are correct and one is wrong. Neither of your interpretations matches that. You need all kinds of additional information to say anything more specific. "99% of results are correct" can easily be true while p(sick | positive) and p(positive | sick) each vary anywhere between 0 and 1.

sykh 8y ago

I updated the problem. Sorry for the mistake.

thaumasiotes 8y ago

> If we follow your explanation, then the probability of having the disease is indeed 99%. This is not correct; the probability of having the disease is unknown. He didn't say what he meant by the test being "99% accurate", but that doesn't mean you can just make your own assumption. Note that in your more precisely specified scenario, when the test has 99% reliability, it is perfectly true that "99% of the time the test gives a correct result", which immediately disproves the claim that, if we follow that definition, the probability of having the disease given a positive test result is 99%.

cdancette 8y ago
The problem is that "99% the time gives a correct result" is imprecise. It can be understood as both: - p(sick|positive) = 0.99 - p(positive|sick) = 0.99 We get totally different results, the first one is obvious (99% change of being sick), and the second one needs Bayes' Theorem (and is the one we want to use).
- thaumasiotes 8y ago
  I would only interpret "the test gives a correct result 99% of the time" to mean that out of every 100 test results, 99 are correct and one is wrong. Neither of your interpretations matches that. You need all kinds of additional information to say anything more specific. "99% of results are correct" can easily be true while p(sick | positive) and p(positive | sick) each vary anywhere between 0 and 1.