A test is developed to detect a certain medical condition. The test is not perfect, and sometimes
gives incorrect results. The behaviour of the test on 1000 randomly selected members of the
population is shown in this tree diagram, where the following notation is used:

• 𝐶𝐶 = has the condition
• 𝐶𝐶 ′ = does not have the condition
• 𝑇𝑇 = tests positive for the condition
• 𝑇𝑇′ = tests negative for the condition
  Three of the branches’ proportions are shown in the tree diagram:
• 1
  100 of the 1000 people have the condition
• 4
  5 of those with the condition test positive for the condition
• 1
  10 of those without the condition test positive for the condition
  A person is selected at random from these 1000 people, and tests positive for the condition.
  What is the probability that this person has the condition?

Answer:

The following is a partially completed tree diagram, showing all the information we need.
Therefore 8 + 99 = 107 people test positive, of whom 8 have the condition, so the probability that
someone has the condition, given that they test positive, is 8
107.
Comment: This is only about 7.5%, so the test is quite poor on this population: over 90% of the
positive test results are false positives. Tests for rare conditions which are applied indiscriminately
will often suffer from this problem, and can lead to costly treatments (physically, financially and
emotionally) for many perfectly healthy people.
The answer is E