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. :helper

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. :yikes: You're utterly devastated and start planning your last days. :cryer

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

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 :laugh

[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!!! :DOK. 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 :DYeah, 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 testIn 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? :mad:

[Win2Win]

So where do the rabiits come into it? :doh

[vegyjones]

So where do the rabiits come into it? :doh

Are they a new species of Jewish Rabbit? :doh

[bigcumba]

Big C.

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

Is it? :mad:

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!

[mathare]

Sounds like a cracking book though!It is. It also covers hypnosis techniques, memory feats etc and how they are performed.

But learning how the brain/mind works is really quite interesting.

[vegyjones]

Can't believe that no one mentioned my great Jewish Rabbit joke.

Come on people, as proved earlier on in this thread, this is the stuff of genius! :D

[silax]

just got in
i would have got it wrong :ooo

[TheOldhamWhisper]

You can wrap it up in as many word games as you want - the chances are still 99% that the test result is accurate - your test is positive - it is therefore 99% positive - no other factors need to be taken into consideration.

The 'trick of the mind' is trying to get you to take something into account which is irrelevant and then convince you that Vegy is a genius :laugh

[mathare]

You can wrap it up in as many word games as you want - the chances are still 99% that the test result is accurate - your test is positive - it is therefore 99% positive - no other factors need to be taken into consideration.I agree with that. But that doesn't mean you have a 99% chance of having the disease does it?


The 'trick of the mind' is trying to get you to take something into account which is irrelevant and then convince you that Vegy is a genius :laugh:laugh

[TheOldhamWhisper]

I agree with that. But that doesn't mean you have a 99% chance of having the disease does it?

The test is to determine if you have the disease - the result of that test is 99% accurate - it comes back as positive - you can be 99% positive that the result is accurate - ergo 99% certain that you have the disease!

[bigcumba]

The test is to determine if you have the disease - the result of that test is 99% accurate - it comes back as positive - you can be 99% positive that the result is accurate - ergo 99% certain that you have the disease!

You obviously 'see' the question in the same way I did Oldham...

[mathare]

You obviously 'see' the question in the same way I did Oldham...And that's the way I saw it initially, until it was explained to me. Now I 'see' it the other way.

[bigcumba]

Just goes to show how differently our brains can work. I will have to get myself a copy of the book anyway as it sounds very interesting.

[mathare]

Just goes to show how differently our brains can work. I will have to get myself a copy of the book anyway as it sounds very interesting.Absolutely.

The bit on how seances and ouija boards work is very interesting too.

[TheOldhamWhisper]

Ok, I have a 10,000 side dice and the chances of me rolling a 1 are 9999/1.

I have written a computer program that predicts to a 99% accuracy the result of the aforementioned dice roll. The computer predicts that a 1 will be rolled. The 'mind trick' is based on asking what are the chances of the dice roll being a 1.

There are only 2 ways to determine the answer - either the chances are 1 in 10000 as stated in the first sentence (in italics) OR you disregard the sentence and use the other fact which suggests that the program is 99% accurate and has predicted that a 1 will be rolled.

You can't mix and match the statements - it is either/or.

[TheOldhamWhisper]

Additional:

My explanations probably won't sell any books :laugh

[mathare]

Ok, I have a 10,000 side dice and the chances of me rolling a 1 are 9999/1.

I have written a computer program that predicts to a 99% accuracy the result of the aforementioned dice roll. The computer predicts that a 1 will be rolled. The 'mind trick' is based on asking what are the chances of the dice roll being a 1.

There are only 2 ways to determine the answer - either the chances are 1 in 10000 as stated in the first sentence (in italics) OR you disregard the sentence and use the other fact which suggests that the program is 99% accurate and has predicted that a 1 will be rolled.

You can't mix and match the statements - it is either/or.OK. Let's have a think about this. One side is 1, the other 9,999 are not. I'm good with that. A 99% accurate computer program says whether or not a 1 will be rolled. Presumably this 99% accurate program will be equally accurate for 'not 1s' as it is for '1s'. We run the program and it predicts a 1.

At this point I go back to your statement that the chance is either 1 in 10,000 (first sentence) or 99% (second part of the puzzle). At least that is how I am interpreting what you've said. Both cannot be true - that discrepancy is huge. So which is right, if either are indeed correct?

The statements, and hence the associated figures should be treated on an either/or basis if they are full independent of one another. Are they? That's what I am trying to rationalise now. Leave it with me a few minutes...

[mathare]

Right, here's my thinking.

There is no independence in those statements. How is the computer program 99% accurate unless it has some form of reliance on the die or it's properties/history?

Sure, the computer could get 99 out of 100 throws of the die correct but it could be fluke. It could most likely get that level of accuracy by predicting 0 (i.e. not 1) each and every time it is asked to predict. Although then it is more likely to be 100% accurate on a sample of 100 throws.

But assuming we are not talking about 100 throws, we're talking about a program that has been certified to 99% accuracy over a statistically significant number of throws then there must be some correlation between computer program and die.

And if there is dependence in the statements then the probabilities are also co-dependent and cannot be treated separately.

[TheOldhamWhisper]

Ok, I've rewritten the computer program and now it produces the same 99% accuracy. This time, it predicts the dice roll will be a 9. What are the chances that the dice roll will be a 9?

(One thing you may wish to consider is that the dice faces may not all be different - something that may or may not have relevance) :D

[mathare]

Ok, I've rewritten the computer program and now it produces the same 99% accuracy. This time, it predicts the dice roll will be a 9. What are the chances that the dice roll will be a 9?

(One thing you may wish to consider is that the dice faces may not all be different - something that may or may not have relevance) :DSurely the similarity of the faces or otherwise is irrelevant. We are simply looking at 9/not 9. As long as the probability of the even happening hasn't changed then this is still the boolean true/false, yes/no, positive/negative consideration. So does it matter whether the number is 1, 9 or pi?

If we look at the chances of the next roll NOT being a 9 then we must be able to then affirm what the chances of it being a 9 are.

A 99% accurate computer has said that a 1 in 10,000 event will happen. Will it? The computer is wrong 1% of the time. 1 in every 100 tries the computer will say 'not 9' and the result will be 9. Also 1 in every 100 ties the computer will say 9 and the result will be 'not 9'.

Now my old brain wiring is trying to get me to say 99% and get on with something else but I refuse to :laugh. Rational thinking to the fore, please.

There are four possible outcomes:
1) Computer predicts 9, dice is 9
2) Computer predicts 9, dice is 'not 9'
3) Computer predicts 'not 9', dice is 'not 9'
4) Computer predicts 'not 9', dice is 9

The computer is right in two cases and wrong in two cases. Now let's look at the probabilities of each event
1) 99/100, 1/10000
2) 1/100, 9999/10000
3) 99/100, 9999/10000
4) 1/100, 1/10000

I paused heavily before writing those. I asked myself what is the probability of the computer predicting a 9, and the answer is "I don't know, I haven't been told that piece of information." At least not directly. Do I know it indirectly? Am I going to kept awake by this all night? :wink

What I have used is the probability of the computer being correct based on knowledge of the die value. For example, in case 1 IF the die comes up 9 then it's a 99% chance the computer also said 9.

But think about it another way. The computer has a 99% chance of predicting each outcome correctly. So it has a (99/100)/10000 chance of predicting a 9, doesn't it? Or does it, because sometimes it will predict a 9 when it 'shouldn't'. So is it once more just 1/10000, the same as the die has of coming up 9.

Argh. :23_111_9[

A few minutes to clear my head and gather my thoughts if I may...

[mathare]

Ok, I've rewritten the computer program and now it produces the same 99% accuracy. This time, it predicts the dice roll will be a 9. What are the chances that the dice roll will be a 9?

(One thing you may wish to consider is that the dice faces may not all be different - something that may or may not have relevance) :DOK, let's spin this round a little. What are the chances that the dice will roll a 9? And why?

[mathare]

I'm back (but so is Vegy I see :yikes:). Anyway...

1 time in 10,000 the die will come up 9 regardless of what your computer says. That means 9,999 times 9 won't come up.

The computer has predicted 9 and 99% of the time it will be right; 1% of the time it will be wrong.

So the computer has wrongly predicted a 9 in 1% of 9,999 (99.99) cases and correctly predicted a 9 in 99% of 1 case (0.99). So the chances of the die coming up 9 IF the computer has predicted 9 are 0.99/99.99 = 0.99%. No?

We are definitely interested in the chance of one event happening given that a related event has already occurred. In this case the chance that a 9 will be rolled having already seen the computer (which I assert cannot be independent) predict it.

[vegyjones]

Glad we cleared that one up :)

[mathare]

Glad we cleared that one up :)You've nothing to add then Jonesy? Don't want to voice your opinions on the matter? :wink

[vegyjones]

No, you and oldham took the words right out my mouth :D

[mathare]

No, you and oldham took the words right out my mouth :DBoth of us? How balanced of you to see both sides of the debate. And how cowardly of you to agree with both of them :D

[TheOldhamWhisper]

Ok, I've rewritten the computer program and now it produces the same 99% accuracy. This time, it predicts the dice roll will be a 9. What are the chances that the dice roll will be a 9?

(One thing you may wish to consider is that the dice faces may not all be different - something that may or may not have relevance) :D

Notice here that the 'rules' have changed - We still know the chances of rolling a 1 but the statement in panetheses now allows for multiple faces with the number 9. The question is how relevant is this fact?

We now have too many unknowns to make a mathematics based prediction - the only solid piece of 'evidence' is that the computer program is 99% correct.

This could go on for a VERY long time and both arguments are valid - there cannot be a definitive answer (which is why this kind of 'trick' is so popular) and I think we can agree on that :)

Excellent debate, Mathare - I'll look forward to the next one :wink

[vegyjones]

I 'm still ebbing and erring :D

[mathare]

Notice here that the 'rules' have changed - We still know the chances of rolling a 1 but the statement in panetheses now allows for multiple faces with the number 9. The question is how relevant is this fact?I did mention in my reply to this that I had assumed the probability of the chosen number coming up had not changed. And continued on that basis. You're correct to state that as written the probability MAY have changed but we don't know either way. And as you say...

We now have too many unknowns to make a mathematics based prediction - the only solid piece of 'evidence' is that the computer program is 99% correct.


This could go on for a VERY long time and both arguments are valid - there cannot be a definitive answer (which is why this kind of 'trick' is so popular) and I think we can agree on that :)Absolutely. It soon ventures outside the realm of maths and statistics and rational thought and reasoning get applied, sometimes badly, to often override the maths that has gone before. Like the voice in the back of my head wanting me to just accept 99% as the answer earlier.


Excellent debate, Mathare - I'll look forward to the next one :winkBeen a pleasure :) And I know this will keep me awake later till I can convince myself one way or the other. :laugh

[GlosRFC]

This is similar to the mathematical conundrum that
0.9999 recurring = 1

[TheOldhamWhisper]

Suppose 1,000,000 take the test. Of those 10,000 will have the disease....

Just to help you sleep - if 1 in 10000 will get the disease, your maths is fundamentally flawed :wink

[TheOldhamWhisper]

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.



Sorry mate - as the computers used to say - invalid argument! You changed the sample size - The sample size is 1 million. :D

(I'm going to bed now - sweet dreams) :ooo

[mathare]

Sorry mate - as the computers used to say - invalid argument! You changed the sample size - The sample size is 1 million. :D

(I'm going to bed now - sweet dreams) :oooThat'll teach me to try and type the explanations out while trying to do some work also. I was much more focused during the latter part of the debate which is when I was at home, thinking in my own time.

You're just being pedantic anyway. But that's OK. I know you're not picking my arguments apart but for the sake of sanity you're trying to prove Vegy isn't a genius :D

[MarcusMel]

This is similar to the mathematical conundrum that
0.9999 recurring = 1

Depends if you are an Applied mathematician or a Pure Mathematician

(engineers rule!)

[vegyjones]

Just so you know,
I'm coming down on Oldham's and BigC's way of thinking now.

A 99% acurate test to determine that you have the disease
is totally independant of the theory that a 99% accurate test means 1 out of 100 tests will be wrong!

[mathare]

Just so you know,
I'm coming down on Oldham's and BigC's way of thinking now.

A 99% acurate test to determine that you have the disease
is totally independant of the theory that a 99% accurate test means 1 out of 100 tests will be wrong!But it is fully independent of the fact that it only affects 1 in 10,000?

[vegyjones]

Yes.

The 99% accuracy suggests that if you ran the test on one person, it would come back positive 99 times.

Doesn't account for tests on 100 different people.

That's the way it's coming to me in my genial brain at the moment :laugh

[TheOldhamWhisper]

If you want, I can provide convincing (if not categorical) statistical proof that I was completely wrong with my arguments :laugh

I think we can all agree that whilst Vegy may not be a genius afterall, he is certainly a Master Debater :D

[mathare]

If you want, I can provide convincing (if not categorical) statistical proof that I was completely wrong with my arguments :laughYes please :D


I think we can all agree that whilst Vegy may not be a genius afterall, he is certainly a Master Debater :D:laugh

[vegyjones]

You're all just clearly jealous of my cleveressness!

[TheOldhamWhisper]

Ok, here goes:

The minimum acceptable sample is 1 million (otherwise we cannot get a potential 100 people who have the disease).

Of the 100 people who have the disease, 99 will test positive whilst 1 person will wrongly be diagnosed as negative.

There will be 999,900 people who do not have the disease of which 9999 people will be wrongly diagnosed as positive.

In total, 10098 people test positive out of 1 million people with (potentially) 1010 people wrongly diagnosed (rounded). This leaves a 'true' chance of being diagnosed positive AND actually having the disease as 1,000,000/9088 = 1:110.

The chance of any single test being 99% accurate cannot be used because the sample is too small to produce statistical accuracy.

(Lies, damned lies and..... :laugh )

[vegyjones]

Of the 100 people who have the disease, 99 will test positive whilst 1 person will wrongly be diagnosed as negative.

I thought we'd already decided that these two statements were completely independant of each other and the fact that a test is 99% accurate doesn't mean that out of 100 people, 1 will get a negative result.

[TheOldhamWhisper]

The only facts we are given are: 1 person in 10000 gets the disease and the test is 99% accurate for both positive and negative tests. The statements are mutually exclusive but as they are the only facts we have, we need to extrapolate the data from them.

Given that we need a minimum of 100 people to be able to prove the positive aspect of the second statement, we can deduce that we need a minimum of 1 million people are needed for the sample (as per the first statement). The rest is perfectly accurate (statistically). :)

[GlosRFC]

I don't understand why you're arguing about this....


...if you'd stayed away from Vegy in the first place, you wouldn't have caught the disease!

[TheOldhamWhisper]

...if you'd stayed away from Vegy in the first place, you wouldn't have caught the disease!

:laugh - that gets you some TOW rep!

[vegyjones]

Don't make me hungry.
You won't like me when I'm hungry! :mad:

http://tbn0.google.com/images?q=tbn:zEYeN-ALDWI8GM:http://imagecache2.allposters.com/images

[bigcumba]

The incredible sulk :)