If I know a conditional probability, what happens if I square the event?

I computed the probability if an event E given an event T. Now I want to compute the probability of E given T happened twice independently? How can I go about this? So I'm basically looking for $$P(E|(T_1, T_2).$$ $$P(T_1, T_2)=P(T_1)P(T_2)$$ since they are independent. Now, $$P(E|(T_1, T_2) = \frac{P((T_1, T_2)|E)P(E)}{P(T_1)P(T_2)}$$ should be what I'm looking for, correct? I know every probability but $$P((T_1, T_2)|E).$$ How do I find that out?

• Your basic manipulations are good. There is no way to determine $P[T_1,T_2|E]$ without more information. – Michael Oct 29 '19 at 20:58
• I'm quite confused why that's not possible. I mean, I believe you, but then I must have understood something wrong. I think my professor is not very accurate when it comes to asking questions .. I edited my question, you can now see the problem set (this question is referring to number 2) and how I solved the first part. – Marc Nov 1 '19 at 11:56

You missed an important word. In the problem statement, we are told that the two tests are conditionally independent. That is to say: $$P(T_1 , T_2 | E) = P(T_1 | E) . P(T_2 | E)$$ and: $$P(T_1 , T_2 | \overline E) = P(T_1 | \overline E) . P(T_2 | \overline E)$$ But this is not the same as being independent. And in fact we have: $$P(T_1 , T_2) \neq P(T_1) . P(T_2)$$
• So if I understand correctly, all I have to do now is square the $P(T|E)$ in the solution provided? – Marc Nov 1 '19 at 16:31