Bayes Theorem and Diseases/Machines

Question

Confused on how to apply Bayes Th. to this problem.

You have a machine that can identify a disease in 75% of cases, and in 95% of cases the machine is able to correctly indicate that a person does NOT have a disease. What is the probability of a machine incorrectly saying a person has a disease when they do not, and what is the probability of the machine incorrectly saying a person does not have a disease when they actually do?

MY WORK:

Assume M stands for the machines predictability. Assume D stands for a person having the disease.

P(M)=0.75 which means the machine correctly predicts 75% of the time, P(M')=0.25 which means the machine incorrectly predicts 25% of the time.

P(M|D')=0.95 meaning a person does not have a disease and the prediction is positive.

I need to first find P(M'|D) is the probability of a person having the disease but the prediction is wrong. I then need to find P(M'|D') which is person not having disease and prediction is wrong.

I am stuck here. Is my thought process correct? Do I not need to have information of P(D) to proceed?

Answer 1

I think you are going astray because your interpretation of the problem statement is a bit off. When the problem states:

You have a machine that can identify a disease in 75% of cases

This is conditional on the disease being present, i.e., $P(M|D) = 0.75; P(M'|D) = 0.25.$

Then, you won't need need $P(D)$ to answer the two questions that were posed - they can be determined by thinking through what $P(M'|D')$ and $P(M'|D)$ represent.