4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:

• Test A is known to give a positive result, indicating a presence of the disease, in \(90 \%\) of cases, a negative result in \(2 \%\) of cases and a requirement for the administration of Test B in \(8 \%\) of cases;
• Test B is known to give a positive result in \(98 \%\) of cases and a negative result in 2\% of cases.

For those 60 -year-old men not suffering from the disease:

• Test A is known to give a positive result in \(1 \%\) of cases, a negative result in \(80 \%\) of cases and a requirement for the administration of Test B in 19\% of cases;
• Test B is known to give a positive result in \(1 \%\) of cases and a negative result in \(99 \%\) of cases.
  1. Draw a tree diagram to represent the above information.
  2. 1. Hence, or otherwise, determine the probability that:
        (A) a 60-year-old man, suffering from the disease, tests negative;
        (B) a 60-year-old man, not suffering from the disease, tests positive.
     2. A random sample of ten thousand 60-year-old men is given the screening tests. Calculate, to the nearest 10, the number who you would expect to be given an incorrect diagnosis.
  3. Determine the probability that:
     1. a 60-year-old man suffers from the disease given that the tests provide a positive result;
     2. a 60-year-old man does not suffer from the disease given that the tests provide a negative result.