A conditional probability problem, concerns a disease and a test to check its existence. The test is not totally reliable, i.e the result might be positive, when the person is healthy and vice versa. Consider the following events:
$D$ : The person has the disease
$P$ : The test result is positive
$N$ : The test result is negative
Knowing the values of $\mathbb{P}[D]$ and $\mathbb{P}[P|D]$, I found the following values :
$\mathbb{P}[P], \mathbb{P}[N], \mathbb{P}[D|P], \mathbb{P}[D|N], \mathbb{P}[N|D]$
We are searching the probability that the person has the disease, given that two tests have been done, where the result of the first was positive and the result of the second negative. Also, we know that each test is independent.
I think that $\mathbb{P}[D|P_1 \cap N_2]$ is the requested probability, where $P_1 \cap N_2$ is the event of the first test result being positive and the second negative. We also know that $\mathbb{P}[P_1 \cap N_2]$ = $\mathbb{P}[P]\mathbb{P}[N]$ , due to independence. But, I don't know how to proceed.
Thank you in advance
By Bayes' rule, we have \begin{align} \mathbb{P}(D|P_1 \cap N_2) &= \frac{\mathbb{P}(P_1\cap N_2|D)\mathbb{P}(D)}{\mathbb{P}(P_1 \cap N_2)}\\ &=\frac{\mathbb{P}(P_1|D) \mathbb{P}( N_2|D)\mathbb{P}(D)}{\mathbb{P}(P)\mathbb{P}( N)}\\ \end{align}
Each of these terms have been found by you earlier.