A False-Positive Puzzle rant

Okay, so, I'm going through the book, Introduction to Probability by Dimitri P. Bertsekas and John N. Tsitsiklis, 2nd edition, working through each problem before moving on to the next section. Section 1.4 is the TOTAL PROBABILITY THEOREM AND BAYES' RULE. Coo' coo'

Going along, the last example giving is this:

Example 1.18. The False-Positive Puzzle. A test for a certain rare disease is assumed to be correct 95% of the time: if a person has the disease, the test results are positive with probability 0.95, and if the person does not have the disease, the test results are negative with probability 0.95. A random person drawn from a certain population has probability 0.001 of having the disease. Given that the person just tested positive, what is the probability of having the disease?

If A is the event that the person has the disease, and B is the event that the test results are positive, the desired probability, P(A|B), is

 

P(A|B) = P(A)P(B|A) / P(A)P(B|A) + P(Ac)P(B|Ac)
       = 0.001 · 0.95 / 0.001 · 0.95+0.999 · 0.05
       = 0.0187

Note that even though the test was assumed to be fairly accurate, a person who has tested positive is still very unlikely (less than 2%) to have the disease. According to The Economist (February 20th, 1999), 80% of those questioned at a leading American hospital substantially missed the correct answer to a question of this type. Most of them said that the probability that the person has the disease is 0.95!

Okay, so, when I read this example / problem, I was confused, because the answer is very clearly 0.95, why is this a question?

I worked through the problem and realized where my, and 80% of those questioned at a leading American hospital, confusion lay. The question is terribly poorly worded and misleading.

Given a positive result, event A, is 0.95. When a doctor in a hospital is dealing with people, they are dealing with one person: the patient. They have a sample set of 1. The probability of that person, that sample set, having the disease after a postiive result is 0.95. Those questioned are answering for event A, the single patient.

What the question is trying to ask is "Given that the person just tested positive, what is the probability out of everyone, of this single person having the disease?" They are answering, given we have event A, what is the probability of A in all of event B space? A reasonable question.

However, that "my patient" versus "everyone" is considerably important.

And I am mildly irritated about the ambiguity in this.

It is like "I'm sorry," and getting "It's okay, not your fault." No, I didn't say it was my fault. I said, "I am (sad) sorry about this situation," not "I am (apologizing) sorry for this situation." They are considerably different.