1997 TYS

A certain disease is present in $1$ in $200$ of the population. In a mass screening programme, a quick test for the disease is used but the test is not totally reliable. For someone who does have the disease there is a probability of $0.9$ that the test will prove positive, whereas for someone who does not have the disease, there is a probability of $0.02$ that the test will prove positive.

1. A person is chosen at random and tested.
   
   1. Copy and complete the tree diagram which illustrates the application of one test.
   2. Find the probability that the person has the disease and the test is positive.
   3. Find the probability that the test is negative.
   4. Given that the test is positive, find the probability that the person has the disease.
2. People for whom the test proves positive are recalled and re-tested. Find the probability that a person has the disease if the second test also proves positive.