H2 Maths 6.1 - Conditional Probability via Tree Diagram

Step 1 - Set up the problem

A disease affects two percent of the population.

Step 1 - Set up the problem

A screening test correctly identifies ninety percent of infected people.

Step 1 - Set up the problem

But the test also returns a false positive five percent of the time for healthy people.

Step 1 - Set up the problem

Given a positive result, find the probability that the person actually has the disease.

Step 2 - Label the tree branches

Draw two first-stage branches: disease, with probability 0.02, and no disease, with probability 0.98.

Step 2 - Label the tree branches

From each node, draw two second-stage branches: positive or negative test result.

Step 2 - Label the tree branches

Write the conditional probabilities beside each second-stage branch.

Step 3 - Multiply along branches

Multiply the probabilities along each path to get the joint probability at each leaf.

Step 3 - Multiply along branches

Disease and positive: 0.02 times 0.90 equals 0.018.

Step 3 - Multiply along branches

No disease and positive: 0.98 times 0.05 equals 0.049.

Step 3 - Multiply along branches

The other two leaves cover the negative results.

Step 4 - Find P(+) using the total probability rule

A positive result can come from two paths: disease and positive, or no disease and positive.

Step 4 - Find P(+) using the total probability rule

Add those two joint probabilities to get the total probability of a positive test.

Step 4 - Find P(+) using the total probability rule

0.018 plus 0.049 gives 0.067.

Step 5 - Apply conditional probability and read the result

Now divide $P(D \cap +)$ by $P(+)$.

Step 5 - Apply conditional probability and read the result

0.018 divided by 0.067 gives approximately 0.269.

Step 5 - Apply conditional probability and read the result

So despite a positive result, there is only a 27 percent chance the person has the disease.

Step 5 - Apply conditional probability and read the result

Common pitfall: do not use P of D alone - always condition on the positive test by dividing by P of plus.