0
$\begingroup$

Let S be symptoms, D disease and T a positive test. We have $P(S) = .2, P(D|S) = .5, P(D) = .01, P(T|D) = .99, P(T) = .1, P(ST|D) = .1$ and $P(T|S) = .5$.

I want to find $P(D|ST)$, so the probability someone has the disease given symptoms and a positive test.

From Bayes I get $P(D|ST) = \frac{P(ST|D)P(D)}{P(ST)}$, I also know $P(ST) = P(S) P(T|S)$.

So I then get (.1 * .01) / (.2 * .5) = .01. So a 1% chance you have the disease given S and T.

Is there a fundamental way to derive the same probability when you for instance don't know $P(ST|D)$ but you have to derive it?

$\endgroup$
3

1 Answer 1

Reset to default
0
$\begingroup$

You can derive all these values fundamentally if you have the joint distribution of the random variables S, D, and T. You can then take integrals of the joint PDF (or sums over the joint PMF) to find these values.

Here is a khan academy video that explains the concept in the two variable case.

https://www.khanacademy.org/math/ap-statistics/analyzing-categorical-ap/distributions-two-way-tables/v/marginal-distribution-and-conditional-distribution

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .