This isn't a paradox...

the answer is 50%, and krellin is right, it isn't a paradox, it's just a bit of confusion in the way we interpret the stats.

actually, isn't it 33%? Much more that the 2% you'd think, that's the point.

Real Answer: It doesn't matter what the probability is that you have the disease, because under Obamacare you will be denied coverage, as your life has been deemed unnecessary to the sustaining of the nation. That'll teach you to vote for the wrong guy on your secret ballot, and fail to make the proper political donations...

Did you read that in Russ Kick's book, Maniac?

lol krellin

33% sounds about right, YJ

Maniac's example appears to be an attempt to describe "Simpson's paradox", however I don't think the particular example given is actually a paradox.

99 + 2 = 101 (positives results in 200 test assuming even distribution)
1 + 98 = 99 (negative results)

Eliminaye the negatives and that means there is a 2 in 101 chance of a false positive so the person his a 99 in 101 chance of actualy having the disease.

.99 * .01/(.99 * .01 + .99 * .02) = 1/3

It's just conditional probability. Let A be the event that you have the disease, and B be the event that you test positive.

P(A|B) = P(B|A)P(A) / P(B) = .99 * .01 / (.01 * .99 + .99 * .02) = 1/3

Wouldn't that be .98*.02 or .98*.01?
And explain why? I see that 2 out of 101 positives are false. so I stick by my argument until an actual explanation behind that multiplication and division works out.

Beaten to the bunch, nice uclabb.

But why .99 in all cases?

Where do you get .98?

That's the probability of testing negative if you don't have the disease, but you shouldn't be concerned with that value and should focus solely on probabilities of testing positive in order to keep the events in the conditional calculation consistent.

but .01 is the probability of testing negative if you are positive, is it not? I'm so confused.

Draug- Here is some intuition. Imagine if the probability of having the disease were 1/10000000000. Then your "method" would give about the same result of 100/102, when intuitively I think it is clear that the probability of having the disease is still essentially zero.

Here's the thing, we *know* the person tested positive. 99 out of 102 times, if the person tested positive, they are positive.

It's not about the odds of the disease. It's about the odds of the outcome of a test for the disease.

The odds of contracting the disease could be 1 trillion to 1. But if the test says positive and 99 times out of 101 positive results it is positive, then the odds are you have the disease. Accuracy isn't about actual occurance odds. It is about accuracy. That is the question as presented. The test is 99% accurate in people who have it and 98% accurate in people who don't.

I think reviewing it that it is starting to make sense. The one in three is because the odds of actually having it affect the odds of a false positive, increasing them with the greater long shot.

I have to grep things in real world terms...

Draug- Are you doing the "trolling" thing where you pretend like you don't understand probability like you claimed you were last time, or do you actually not understand? What I am saying is correct. It's not a debate.

So, my number would have been right if the odds of having the disease were exactly 50/50.

That was a cross post. My apologies.

But the point is that if a test is incredibly rare, then that alone implies that a positive result is very likely a false positive rather than an actual positive.

Say you have 1,000,000 people in a room and one of them has a disease. If a test gives false positives for healthy patients 2% of the time, then you're going to get close to 20,000 positive results testing all those people. The odds that any individual one of them has the disease is going to be close to 1/20,000 then. Way more false positives are given out than actual positives if the disease is rare.

Nope, just working through it in my head and I think it make sense now because you all are showing that the odds of actually having the disease affects the adds of a false positive. In this case, a false positive is 99*2 times more likely than a false negative. So you have 99 real positives out of 10,000 cases but 198 false positives so only a 1/3rd chance. Got it.

These probability calculations all require that you know the percentage of the population that should have the disease, which should be noted. You can't calculate these odds without that statistic known or assumed.

Got it sorted, Y2K. My original calc only applies in 50/50 or unknown ratio scenarios.

Put in Y2K's room example.

100 people in the room should result in one actual positive result with the remaining 99 people having a 2% chance at a false positive, so 2 more people should show positive - or a one in three chance of any specific positive result being that one person who actually had it.

But that does lead one to ask if it isn't actually 1 in 2.98 odds (or 50 in 149, slightly less than 1 in 3). I'm sure the 1 in 3 is right because the percentages are based on the total 100 people, not the remaining 99. So out of 100 people there will be 1 with it 2 with false positives and the rest with negative results, one of whom could be the person with it having a false negative, but the odds of that are 1 in 10,000.

@'. In this case, a false positive is 99*2 times more likely than a false negative. So you have 99 real positives out of 10,000 cases but 198 false positives so only a 1/3rd chance. Got it.'

Very good, Both correct and explained in plain english without the mathematical jargon (i don't get the jargon, but can do the calculation in my head, so saying why my answer is right in any language is hard for me)

The point about needing to know the actual percent infected and how to work out the probability of a false positive/negative is valid; but the principle is that a false positive can be much more likely than a true positive even with 99&98% accuracies, hence the paradox, or counter intuitiveness of the result.

In the real world you don't necessarily have the data to make these calculations, beyond estimate. But that doesn't change the princple.

Well, it isn't a paradox, but it is counter intuitive from a certain point of view, making it an apparent paradox, but not an actual one.

I thought this thread would be about something from Paradox Interactive. I'm sad.

This is actually the argument I've seen libertarians use against using metadata for counter terrorism:

In a population of 100 million, 10 thousand of them are planning terrorist plots this year. The government has been tapping the emails of all its citizens and discovered that 99% of terrorists use the word 'bomb' in their emails, while only 1% of civilians do.

The government has the funds to foil the actions of up to 30,000 would be terrorists.

On average, how many of the would be terrorists would actually be followed?
What is the probability that 9,900 terrorists (all of them that used the trigger word) will be followed?

The numbers used above aren't realistic, in reality far fewer than 99% of terrorists will use a trigger word in their emails, metadata is frankly useless for counter terrorism and since it infringes on our privacy rights, we shouldn't use metadata.

"I thought this thread would be about something from Paradox Interactive. I'm sad."
When I first opened it, I did too, lol

"I thought this thread would be about something from Paradox Interactive. I'm sad"
Same with me at first haha

I didn't think it was a true paradox when I read it but I deferred to the author who described it as Simpson's Paradox in a BBC article.

@fasces - the use of this as an argument for or against metadata collection is flawed. If we tested everyone for the disease it is true that the positive results are only reliable 1/3rd of the time, but the negative results are reliable 99% of the time. Using the test to target resources at the positive results catches the vast majority of those infected. For example you may now wish to give an MRI scan to all those tested positive, but you couldn't possibly afford to give MRI scans to everyone. It is true that you will miss 1 in 100 sufferers but you will detect and possibly treat the other 99.