N2012 P2 Q5

The probability that a hospital patient has a particular disease is \(0.001\). A test for the disease has probability \(p\) of giving a positive result when the patient has the disease, and equal probability \(p\) of giving a negative result when the patient does not have the disease. A patient is given the test.

1. Given that \(p=0.995\), find the probability that
   1. the result of the test is positive,[2]
   2. the patient has the disease given that the result of the test is positive.[2]
2. It is given instead that there is a probability of \(0.75\) that the patient has the disease given that the result of the test is positive. Find the value of \(p\), giving your answer correct to 6 decimal places.[3]