The statement:
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.
The problem:
I can solve this problem, but I'm unable to understand what is wrong with the following:
Let $T_i$ 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 $P(T_1, T_2) = P(T_1)P(T_2)$, because the tests are independent.
By law of total probability we know that: $$P(T_1, T_2) = P(T_1, T_2\,|\,D)P(D) + T(T_1, T_2\,|\,D^c)P(D^c)$$ Replacing, and assuming conditional independence given $D$, we have: $$P(T_1, T_2) = 0.95^2*0.01 + 0.05^2*0.99 = 0.0115$$
This is the correct result, but now let's consider that: $$P(T_1, T_2) = P(T_1)P(T_2)$$
We know that $T_1 = T_i$ for all $i$ because of symmetry, so we have $P(T_1, T_2) = P(T_1)^2$. Again, by law of total probability: $$P(T_1) = P(T_1\,|\,D)P(D) + P(T_1|D^c)P(D^c)$$ $$P(T_1) = 0.95*0.01 + 0.05*0.99 \approx 0.059$$ So we have: $$P(T_1, T_2) = P(T_1)^2 \approx 0.059^2 \approx 0.003481$$
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.