Probability of a false negative

[Hmore]

Hello
I was wondering what was the probability of the following:
If a person takes a medical test and receives a negative result and the company that makes the test states the possibility of it being accurate is 90% and then the same test is taken a second time and it is also negative ..what is the probability that a negative result is accurate, asuming the company's statement is correct?

 

[Dr.Peterson]

To be sure we're interpreting this correctly, what is the exact wording of the claim about 90% accuracy? (E.g. does it use words like "sensitivity" or "specificity", or "false positives" or "false negatives"?) That will make a big difference.

There's another reason for hesitancy in answering you: If the reason for errors is random and independent, we could give an answer, but very likely that is not true. For example, there might be something special about you such that the test doesn't work well for you, so that the second test really adds no new information. In that case, the probability might be a matter of the percentage of people in the population for whom it works, rather than the percentage of times it works on one person. Most likely the reality is somewhere in between.

 

[Hmore]

The wording is "about 90 percent specificity and 85 percent sensitivity"

 

[HallsofIvy]

There are two ways such a test can go wrong- returning a negative result for a person who actually has the illness (a "false negative") and returning a positive result for a person who does not have the illness (a "false positive"). Saying the test is "90% accurate" (surely not "90% chance of being accurate"!) does not distinguish between the two. If I really had to do this problem, I would assume that the "90%" applied to both. That is, if a person has the illness there is a 10% chance of a false negative and if a person does not have the illness there is a 10% chance of getting a false positive.

However, as Dr. Peterson said, in order to answer this question, you need to know the probability of a person in the general population actually having the disease.

For example, suppose that 30% of the people have the disease. In a population of 1,000,000 300,000 have the disease. 10% of them, 30,000 will get a false negative. Of the 700,000 who do not have the disease 90%, 630,000 will have a "true" negative. So 630,000+ 30,000= 660,000 who tested negative, 630,000 of them, or 630,000/660,000= .954 or 95.4% do not have the disease and 4.6% do not.

Now, with the 660,000 who tested negative, do that again. I am not going to do that because, if we had chosen some other percentage than 30% of people who have the disease we would get a completely different answer!