What’s the probability the test result is positive?

Question

A clinic offers a reliable test for a rare disease. The disease only affects one in ten thousand people. If one has the disease the test has a $98$% chance of giving a positive result, and if one does not have the disease, the test has only a $1$% chance of giving a positive result.

If a person decides to take the test,

(a) what's the probability the test result is positive? (Hint: let A be the event that a test result is positive, and B be the event that the person has the disease)

(b) if the person's test result is positive, what is the probability that he has the disease?

Using Bayes Theorem I got (a) = 0.010097 or 0.0101 and (b) 0.00971

Not sure why this question is on hold... I have asked plenty similar questions with no problems. This question contains all necessary information for it to be solved...like what? I did give my thoughts and I stated I don't understand how to approach it and needed help to get started....

Answer 1

(Hint: let $A$ be the event that a test result is positive, and $B$ be the event that the person has the disease)

Step 1: identify what you are given, as probabilities(conditional or otherwise) in terms of $A,B$

The disease only affects one in ten thousand people. If one has the disease the test has a $98\%$ chance of giving a positive result, and if one does not have the disease, the test has only a $1\%$ chance of giving a positive result.

$$\begin{align}\mathsf P(B) = & ~0.0001\\[1ex]\mathsf P(A\mid B)= & ~ 0.98\\[1ex] \mathsf P(A\mid \neg B)= & ~ 0.01\end{align}$$

Step 2: identify what you want to find, in terms of the same.

(a) what's the probability the test result is positive?

$$\mathsf P(A)$$

(b) if the person's test result is positive, what is the probability that he has the disease?

$$\mathsf P(B\mid A)$$

Step 3: consider what tools you know that allow you to use what was given (1) to find what you need (2). (Bayes' Rule, Law of Total Probability, et cetera.)

Step 4: do it.