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I am getting confused on what should be a fairly simple concept.

A flu test gives true positives 90% of the time, and true negatives 85% of the time. 5% of people have the flu.

We want to find the probability that a healthy person tests positive two times.

To me I feel we should look at the false positives, people that test positive but are healthy. Which calculates at 100-90 = 10. Therefore the probability of testing positive twice and being healthy is 10% x 10% = 1%.

The results I'm seeing from another source is that we should be calculating the false positive rate from the true negative, i.e 100 - 85 = 15. And then 15% x 15% = 2.25%

This seems back to front to me, could I get clarification on which is actually right?

In theory if we have 100 people who tested positive, then to me it seems 90 of them actually have the virus (90% true positive), and 10 tested positive but don't (10% false positive).

mintteaplease
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  • "10%x2" This is a pretty terrible way of writing $10%\times 10%$ or $(10%)^2$. It looked like you were multiplying by two, not raising to the power of two... – JMoravitz Feb 09 '24 at 15:46
  • For what its worth also... the problem seems to want you to assume that these flu test results are independent of one another... but that is rarely the case in reality. If someone tests positive, it is likely because they have particular antibodies present in their blood stream that the test tries to detect, and those same antibodies will be there every time they are tested... – JMoravitz Feb 09 '24 at 15:48
  • I am multiplying by two, 10 x 2 gives the same as 10^2. – mintteaplease Feb 09 '24 at 16:01
  • $10 \times 2 = 20$... You are not multiplying by two... and you suggesting otherwise suggests that perhaps english is not your first language and you may be better helped by someone speaking the same language as you. – JMoravitz Feb 09 '24 at 16:02
  • Apologies you are correct - it is a Friday. However, I am glad for the insight that struggling at math may imply not having English as my first language, that is a new one. – mintteaplease Feb 09 '24 at 16:05
  • $10 \times 2 = \underbrace{10 + 10}{2\ times} = 10 + 10 = 20$. But $10^2= \underbrace{10 \times 10}{2\ times} = 10 \times 10 =100$. These are not the same at all and it's a very basic concept. In english we say $10 \times 2$ is "multiplying by $2$" and we might say $10^2=10\times 10$ we are "multiplying two times". The language and meaning might be confusing but the concepts should be very clear. – fleablood Feb 09 '24 at 16:07
  • For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – Henrik supports the community Feb 09 '24 at 16:23
  • The language is confusing but the author is saying "Sick people test positive 90% of the time; healthy people test negative 85% of the time". If you consider it that way it is easy to say a probability of a healthy person testing positive twice would be $(1-.85)^2=.0225$ I guess it could be interpreted as "90 percent of the positives are true and 85% of the negatives are true" but if 5% are actually sick, those numbers just might not add up. (Or maybe they do). In this interpretation the false positive would be neither 10% nor 15% – fleablood Feb 09 '24 at 16:23
  • Given a positive test there's only a 24% chance the person is sick. Makes this kind of thing counterintuitive. – TurlocTheRed Feb 09 '24 at 17:18

1 Answers1

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As for the numbers involved... $90\%$ is the probability that a person tests positive given that they are truly sick. That is, $\Pr(\text{positive}\mid\text{is sick})$

On the other hand, $100\%-90\% = 10\%$ is the probability that a person tests negative given that they are truly sick... that is $$1-\Pr(\text{positive}\mid\text{is sick}) = \Pr(\neg \text{positive}\mid\text{is sick})=\Pr(\text{negative}\mid\text{is sick})$$ Again, this $10\%$ value is the probability of a false negative. It is not, as you thought, the probability of testing positive given not sick.

Reworded... $\Pr(A\mid B)+\Pr(\neg A\mid B)=1$ is a valid identity. You tried to say that $\Pr(A\mid B)+\Pr(A\mid\neg B)=1$ which is false and invalid.

JMoravitz
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  • Hmm do you know any videos that explain this clearly? I am really struggling to understand why this is the case and not the other way around – mintteaplease Feb 09 '24 at 15:59
  • I am struggling to understand how you could have thought it was the other way around. Rephrase the information in the problem formally as conditional probabilities... you get $\Pr(\text{positive}\mid\text{is sick})$ as $90%$. As for why the identity is as I write... $\Pr(A\mid B)+\Pr(\neg A\mid B) = 1$... that follows trivially from the law of total probability and the definition of conditional probability. Hardly something that needs a whole video... it is at most a one or two line mention in a footnote of any standard textbook. – JMoravitz Feb 09 '24 at 16:06
  • All it requires is noting that $(A\cap B)\cup (A^c\cap B) = B$ – JMoravitz Feb 09 '24 at 16:07
  • Thanks I have being looking at the concepts within textbooks, and yes maybe it means I need to go back to earlier topics. But I usually fine it easier to learn via audiovisual means. – mintteaplease Feb 09 '24 at 16:08
  • @JMoravitz I thought it was the other way around at first. That 90% of the positive test were true (and 10% of the positive tests were false) and 85% of the negative test were true. But that wouldn't tell us anything about how frequently a sick or healthy person tests one way or another unless we also know to total proportion of positive tests to negative tests. – fleablood Feb 09 '24 at 16:32
  • @fleablood That would be a particularly uncommon way of presenting the data. Talking about efficacy of tests in the way I have interpreted them is industry standard and math textbook standard. I have never seen such data presented in any other way. Could the author have made it less ambiguous by including a few extra words? Sure, but from context I do not believe it to have been necessary. – JMoravitz Feb 09 '24 at 16:37
  • "Thanks I have being looking at the concepts within textbooks, and yes maybe it means I need to go back to earlier topics. But I usually fine it easier to learn via audiovisual means. " Honestly I don't feel you difficulty is mathematics but assumptions of language and interpretation. To be fair, I was confused by this at first too but by my interpretation the question was unanswerable. We know that 10% of the positive test go to healthy people and 85% of the neg to healthy people but unless we know the proportion of pos to neg test we don't know how many healthy people test positive. – fleablood Feb 09 '24 at 16:39
  • @JMoravitz that is true and I quickly saw such an interpretation would be useless. But statistics is maybe one of my lesser experiences and I was going from straight language interpretation which... well... we all know where language can lead you in math. On the other hand, a good text would have explained the language clearly from the very beginning. – fleablood Feb 09 '24 at 16:44