# 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 = 0.0187Note 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.

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