Conditional Probability and Independence nonsense in a problem
Suppose that a patient tests positive for a disease affecting 1% of the population. For a patient who has the disease, there is a 95% chance of testing positive, and for a patient who doesn't has the disease, there is a 95% chance of testing negative. The patient gets a second, independent, test done, and again tests positive. Find the probability that the patient has the disease.
I can solve this problem, but I'm unable to understand what is wrong with the following:
Let be the event that the patient tests positive in the i-th test, and let D be the event that the patient has the disease.
The problem says that , because the tests are independent.
By law of total probability we know that:
Replacing, and assuming conditional independence given D, we have:
This is the correct result, but now let's consider that:
We know that for all i because of symmetry, so we have . Again, by law of total probability:
So we have:
The second approach is wrong, but it seems legitimate to me, and I'm unable to find what's wrong.
Thank's for your help, you make self studying easier.