Posterior Probability - why does evidence affect probability? Suppose there a medical test is administered to test if a person has a particular disease:

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*If they have the disease, there is 10% probability that the test says they don't have the disease. This is called false negatives.

*If they don't have the disease, there is 30% probability that the test says they have the disease. This is called false positives.

Suppose that a random patient is given this test. If the test result is positive, what is the probability they have the disease?
Logically, it goes like this
Pr(Positive,Positive) = 100% - False Positive = 100% - 30% = 70%.
Suppose that now it is known that the disease only occurs in 10% of the population. Using posterior probability, the probability is
Pr(Positive,Positive) = 25%
Why does knowing that the disease only occurs in 10% of population change the probability that the patient has the disease? I'm confused; can someone please help me clear my confusion?
 A: The way to frame and interpret medical tests in general is to understand them as updating one's level of certainty that the patient has the disease:

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*without a medical-test result, the disease prevalence (a measure of disease frequency) can be taken as the patient's probability of having the disease;

*however, in the context of a medical-test result, the aforementioned probability has changed: its updated value depends not just on the disease prevalence (as before), but now also on the test's sensitivity (true positive rate) and specificity (true negative rate).

Given a positive test result, the (updated) probability $P(D|+)$ that the patient is indeed diseased can be derived from the following probability tree:

p:  disease prevalance and other (prior) risk factors
v:  test sensitivity
f:  test specificity
D:  Diseased
H:  Healthy
\begin{aligned}P(D|+)&=\frac{P(D+)}{P(D+)+P(H+)}\\&=\frac{pv}{pv+(1-p)(1-f)}.\end{aligned} This formula makes clear that $P(D|+)$ is a function of disease prevalence $p,$ test sensitivity $v,$ and test specificity $f.$
It makes sense that information about the test's technical characteristics ($v$ and $f$), as well as the disease prevalence and the patient's prior health ($d$), should refine our knowledge of the probability that the patient has the disease.
Addendum
OP: Does fewer people having the disease increase the probability that the test result is just a false positive?
Yes. From the above probability tree, the probability that the test result is a false positive is $$P(H+)=(1-p)(1-f).$$ So, the lower the disease prevalence $p,$ the greater this probability; in fact, unless the test has 100% specificity, the number of false-positive results is directly proportional to $(1-p).$
