True that many doctors don't understand the maths (even though it's not actually very hard maths) but what they/we do every day (or should do) is to assess the patient first and try to come to a differential diagnosis of what could be wrong with them.

By doing this, we increase the prior probability that they have the condition we are testing for before we test for it.

So Bayes tells us that a good doctor is worth his or her salt and shows mathematically why this is so.

On the other hand, screening for diseases in the general population is fraught with problems because of low prior probabilities. Most leading physicians that determine whether a disease is or is not worth screening for are quite familiar with Bayes' theorem which is why there are so few national screening programmes.

That really helped working through the question in basic steps thank you. That is the "intuitive Bayes" with the table which I understand a little better than the general Bayes theorem. I would love if someone could show me how to plug those numbers into the general Bayes formula as Dason advised that I should get a handle on both ways of doing this...

(.95)(.002)/[(.95)(.002)]+[(.15)(.998)]=.0125 using General Bayes Theorem right Dason??

Now could you explain to me how the spot where I put the .15 is supposed to be the complement of A (A is .95) why is it 1-.85 and why the spot where I put the .998 is supposed to be B given the complement of A why is it 1-.002?

This is the problem I am having...rationalizing why the numbers go where they do...

I appreciate all the help everyone has given getting me to the correct answer; now I would like to gain some understanding so my next question I don't have to bother everyone again....

I think the table helps build the intuition but you should still be able to do the symbol manipulations and use the given probabilities directly to find an answer too. I'll let H+ be the event that a person actually has the disease, H- be the event that they don't. Let T+ be the event that they test positive for the disease and T- be the event that they test negative.

So putting the information in the problem into probability statements we have that .002 of the people of interest have the disease. The gives us P(H+) = .002 so correspondingly P(H-) = .998

Our sensitivity is .95 which is the probability of a positive test given that the person truly has the disease. This implies P(T+ | H+) = .95

Our specificity is .85 which is the probability of a negative test given that the person doesn't have the disease. This implies P(T- | H-) = .85. Note that P(T-|H-) + P(T+|H-) = 1 (Do you see why this should be true) which implies that P(T-|H-) = 1-.85 = .15

Plugging in the numbers should give the answer. Try to figure out where you were making your mistake before. Hint: You were confusing T and H in the denominator.

Note: Yes I just copied my post from the previous thread and updated the numbers.

And then one last question Dason and I PROMISE you will be rid of me on this question....
When you read a word problem like this how do you decide that Bayes is the route to take and how do you decide the correct events to represent your A and B? That is a major issue I am having as a beginner here. I can't figure out what 2 events will be my "main" letter representatives.

Bayes theorem is useful to flip the conditions around in a conditional probability statement. Say you want to know the probability of X given Y but you don't know it. If you had Y given X though (along with some additional information) you have a chance of figuring out that probability.

In the problems you're dealing with it's easy for somebody to find the probability that a test will return positive or negative if they know whether the disease is present or not.

But that's not the interesting part because what we really want to know is the probability that a certain person has the disease. If we know the results of their test then we're looking for P(Disease | test was positive). We don't have that but there is a lot of information about the P(Test is positive | Disease). So we're looking to flip that condition around. We can't just directly do that. We need some additional information and Bayes theorem tells us exactly what we need to know and how to use that information to get the quantity of interest.