A little maths problem

[mathare]

And I'm expecting good things from you here Silax... :)

Suppose there is a disease going round that affects 1 in 10,000 of the population. For argument's sake let us suppose that everyone is equally likely to get the disease and that anyone catching the disease will die within a week.

A media storm kicks up about the disease and you are getting concerned so you go to your GP. You take a test that is 99% accurate in terms of both positive and negative results, i.e. any result is 99% likely to be correct.

You wait for your results to come back. The test comes back positive. You're utterly devastated and start planning your last days.

But are you right to be that distraught? Estimate your chances of having the disease.

I'll leave this for a while and then post the answer

NB Anyone who has read the book I am currently reading will know the answer to this already. I won't reveal the title of the book just yet but it's a mainstream non-fiction book.

[mathare]

Just as a short aside I gave my missus this same problem. She did as asked and then when told the correct answer argued with me for ages about how no disease can equally affect all the population etc etc. And you how can't trust such medical figures and disease rates going on about 1 in 3 getting cancer but that some people are more predisposed (although she kept saying predispossessed but I knew what she meant) to getting cancer either for genetic or lifestyle reasons.

I reposed the problem replacing the chance of death through this disease with the chance of getting a nice cream bun but she still wasn't having it

[Win2Win]

I only exchange fluids with sheep.....so that counts me out :)

On a side note.....scientists last month brought back to life a 5 million year old virus they'd found!! Luckily it turned out to be weak, and unlikely to damage any human cells. Idiots.

[mathare]

I only exchange fluids with sheep.....so that counts me out :)

Maybe the sheep are the carriers. Maybe the disease is airborne...

On a side note.....scientists last month brought back to life a 5 million year old virus they'd found!! Luckily it turned out to be weak, and unlikely to damage any human cells. Idiots.

:splapme

[vegyjones]

your chances of having it are 99% aren't they.
That's what the result of the test says!!! :D

[mathare]

your chances of having it are 99% aren't they.
That's what the result of the test says!!! :D

OK. Fair enough. If that's what you think.

[bigcumba]

From my point of view, working in the lab, the important bit is that the result is positive, and it's 99% accurate, so the chance of you having the disease is 99%. The 1 in 10,000 bit doesn't enter into the calculation at this time as you're already 99% proven positive.

But.... if you are looking at things before doing the test, you have a 1 in 10,000 chance of having that disease, and the blood test doesn't come into the equation - it can only confirm with 99% accuracy whether you do or not.

[vegyjones]

Right, trhinking about this.

1 in every 100 people will get positive results won't they.
If the disease effects 1 in 10000 people, thyat means if they all got tested 100 poeple would test positive.

Dopes that mean it is still something like 1 in a hundred. or a 1% chance!

[MattR]

0.000099 % ?

[bigcumba]

0.000099 % ?

The thing is the blood test has said you are positive for this disease, and it's only a 1% chance of that being wrong - so it follows that the chances of you having the disease are 99% under those circumstances. The 1 in 10,000 figure is a red herring as you have already been told it's 99% certain you have the disease.

[mathare]

Right, trhinking about this.

1 in every 100 people will get positive results won't they.
If the disease effects 1 in 10000 people, thyat means if they all got tested 100 poeple would test positive.

Dopes that mean it is still something like 1 in a hundred. or a 1% chance!

Bingo!

[mathare]

The thing is the blood test has said you are positive for this disease, and it's only a 1% chance of that being wrong - so it follows that the chances of you having the disease are 99% under those circumstances. The 1 in 10,000 figure is a red herring as you have already been told it's 99% certain you have the disease.

It's not a red herring.

Suppose 1,000,000 take the test. Of those 10,000 will have the disease. That means 990,000 don't have the disease.

Of those that do have the disease (10,000) 99% will be correctly diagnosed as having the disease so that's 9,900 correctly diagnosed as having the disease and 100 incorrectly diagnosed as not having it.

Of those that don't have the disease (990,000) 99% will be correctly diagnosed as being free from disease but 1% will be incorrectly diagnosed and will get a positive test result. That's 9,900 WRONGLY given a positive result.

So if you get a positive result your chances are 99% that it is wrong and 1% that it is right.

[vegyjones]

By jove. I'm a genius :D

[mathare]

By jove. I'm a genius :D

Yeah, who'd have thought it, eh?

[bigcumba]

It's the wording of the question that's the problem for me then Mat... At work if I do a blood test on someone that comes out positive for whatever disease, and the test is 99% accurate, then no matter what the odds were of that person having the disease before the test was taken, the chances of them having it now are 99%. If I was to say that a positive result from a 99% accurate test meant a patient had a 1% chance of having that disease I'd be laughed out of my job!

You take a test that is 99% accurate in terms of both positive and negative results, i.e. any result is 99% likely to be correct.

You wait for your results to come back. The test comes back positive.

Try it another way, just for the purposes of this - you have a 1 in 10,000 chance of carrying the HIV virus having come into contact with it. Your chances of having it are 1 in 10,000. You go for a blood test. It's positive for HIV and that test is 99% accurate. So under those circumstances what are your chances of having HIV. It's now 99%, because you've had a highly accurate test to prove it. It's all about the timing - before the test vs after the test

[mathare]

Finger trouble a minute ago BigC? I saw that reply about a dozen times

[bigcumba]

Finger trouble a minute ago BigC? I saw that reply about a dozen times

Yeah, I don't know what happened there!

[mathare]

It's the wording of the question that's the problem for me then Mat... At work if I do a blood test on someone that comes out positive for whatever disease, and the test is 99% accurate, then no matter what the odds were of that person having the disease before the test was taken, the chances of them having it now are 99%. If I was to say that a positive result from a 99% accurate test meant a patient had a 1% chance of having that disease I'd be laughed out of my job!

Most medical tests are not of the type described in the problem though. The chances of reporting a false positive are usually not the same as reporting a false negative, as far as I am aware. In this test they are the same. And then due to the low chance of the disease being present you get a lot more false positives which means the chance of your positive being a false one is greater.

In reality you're unlikely to be subject to this sort of blood test (or any sort of test) unless you had reason to suspect you may have contracted the disease. Then the results are skewed because only those who have displayed some form of possible symptom take the test.

This is why I tried to talk my missus through the problem using cream cakes rather than medical testing. Were everyone entered into a two-stage lottery (1 in 10,000 go through to the next stage where you're told with 99% accuracy whether or not you have won) then it may be a fairer, less contraversial example. Perhaps.

Try it another way, just for the purposes of this - you have a 1 in 10,000 chance of carrying the HIV virus having come into contact with it. Your chances of having it are 1 in 10,000. You go for a blood test. It's positive for HIV and that test is 99% accurate. So under those circumstances what are your chances of having HIV. It's now 99%, because you've had a highly accurate test to prove it. It's all about the timing - before the test vs after the test

In this example have I already come into contact with the virus? If so then I know that and I am likely to be suspicious as your infection rate accounts only for those who have come into contact with the disease so we know how it is transmitted, at least to some degree. Suppose now for the given example of my disease that it is no longer a disease but a genetic mutation or something that cannot be transmitted but you either have or have not got. The premise is that everyone is equally at risk which is never really the case in medical matters.

I wouldn't have an HIV test now because as far as I know I have never done anything to put me at risk on contracting the virus nor have I displayed any symptoms of it. But were there something else that placed every man, woman and child at the same risk (which HIV doesn't) then I may consider taking a test to see where I stand.

[bigcumba]

I think we're going to have to agree to differ on this one Mat! I can see where you're coming from, but nearly 30 years of doing things a certain way at work have probably left me rather set in my way of looking at this sort of thing!

Maybe if you'd used beer instead of cream cakes... :)

[vegyjones]

Big C.

Is your problem that you just don't want to admit that I'm a genius!

Is it?

[Win2Win]

So where do the rabiits come into it?

[vegyjones]

So where do the rabiits come into it?

Are they a new species of Jewish Rabbit?

[bigcumba]

Big C.

Is your problem that you just don't want to admit that I'm a genius!

Is it?

erm.... well... OK, you got me sussed. :)

[mathare]

I think we're going to have to agree to differ on this one Mat! I can see where you're coming from, but nearly 30 years of doing things a certain way at work have probably left me rather set in my way of looking at this sort of thing!

Aye, that's cool. The major difference I think is whether some sort of pre-test filtering has occurred though. In your line of work it has (why refer someone for an unnecessary blood test unless you have reason to suspect something is awry?); in the problem given it hasn't.

The use of medical testing in the example is deliberate as it is an emotional thing for many so the brain stops thinking in pure mathematical terms and starts to cloud the logic with emotion.

It was taken from the Derren Brown book, 'Tricks of the Mind' I think it's called. It's a fascinating (to me anyway) look at how the human mind works and how we think about things relating to science and maths such as probabilities.

[bigcumba]

. In your line of work it has (why refer someone for an unnecessary blood test unless you have reason to suspect something is awry?);

A question we keep asking Mat, as the workload goes up and up and up... sadly in these days of litigation for anything and everything, they have to cover their asses by screening for tons of things 'just in case' - but they call it 'prevention'. Then you have your 'Well Woman' and 'Well Man' clinics plus all the others, and they wonder why the NHS is skint...

Sounds like a cracking book though!