I'm quite happy to answer part 1. of the following problem.
"You are diagnosed with an uncommon disease. You know that there is only a 1% chance of getting it. Let D stand for the event 'you have the disease' and T stand for 'the test says so'. It is known that the test is imperfect:

and

- Given that you test positive, what is the probability that you really have the disease?

- You obtain a second opinion: an independent repetition of the test. You test positive again. Given this, what is the probability that you really have the disease?"

So, for part 1. I use Bayes' theorem to getwhich is correct, according to my book. However, when I apply similar reasoning to the second scenario I get an impossible answer.This is where I get stuck. I know that the independence of and doesn't necessarily imply their conditional independence (conditioned on ) but I have no idea how to proceed. If I just assume the conditional independence I get an answer greater than 1, which is obviously wrong.