a. To draw a tree diagram for this scenario, we can follow these steps:
Step 1: Start by drawing a root node at the top of the diagram, representing the initial selection of a person.
Step 2: From the root node, draw two branches representing the two possible outcomes: having the disease (D) or not having the disease (D').
Step 3: From each branch, draw two more branches representing the possible outcomes of the test: positive (P) or negative (P').
Step 4: Label the probabilities on each branch. For the having disease branch, the probability is 1/200 (1 in 200). For the not having disease branch, the probability is 199/200 (199 in 200). For the positive test outcome branches, the probabilities are 0.9 and 0.02, respectively. For the negative test outcome branches, the remaining probabilities can be calculated as 1 minus the corresponding positive test outcome probabilities.
The tree diagram should now display all the possible outcomes and their associated probabilities.
b. To find the probability that the person has the disease and the test is positive, we need to consider the two branches that lead to a positive test outcome: having the disease and not having the disease, multiplied by their respective probabilities. So the probability can be calculated as:
Probability(Disease and Positive Test) = Probability(Disease) * Probability(Positive Test | Disease) = (1/200) * 0.9
c. To find the probability that the test is negative, we need to consider the two branches that lead to a negative test outcome: having the disease and not having the disease, multiplied by their respective probabilities. So the probability can be calculated as:
Probability(Negative Test) = Probability(Disease) * Probability(Negative Test | Disease) + Probability(Not Having Disease) * Probability(Negative Test | Not Having Disease) = (1/200) * (1 - 0.9) + (199/200) * (1 - 0.02)
d. Given that the test is positive, we need to find the probability that the person has the disease. This can be calculated using Bayes' theorem:
Probability(Disease | Positive Test) = (Probability(Disease) * Probability(Positive Test | Disease)) / Probability(Positive Test)
We already have the values for Probability(Disease), Probability(Positive Test | Disease), and Probability(Positive Test) from the previous calculations.