Harder Base Rate Question

[lawtonfogle]

Well, you guys so far did great on the first one, so here is the second one.

We know that 100 out of the 10,000 people tested have cancer.

We also know that the test, if you actually do have cancer, is 99% correct.
We also know that the test, if you don't have cancer, will give a false positive 1% of the time.

Bob (same Bob from before), just received a positive result on this test. What is the chance that Bob has cancer?

 

[Ectezus]

Well, you guys so far did great on the first one, so here is the second one.

We know that 100 out of the 10,000 people tested have cancer.

We also know that the test, if you actually do have cancer, is 99% correct.
We also know that the test, if you don't have cancer, will give a false positive 1% of the time.

Bob (same Bob from before), just received a positive result on this test. What is the chance that Bob has cancer?

50%!!!
Either he has, or has not got cancer!


Serious though I would assume: 100 - (1 false positive) * 99% = 98.01.

Although I'm not sure if you mean that 1% false positive is in- or excluded in the 99% correctness.
It depends on: "If you actually do have cancer". Do you mean actually or actually according to the (incorrect) test?

 

[jayem]

50% is correct by my calculations.

I believe what you're asking for is Positive Predictive Value. The formula is:

PPV = (sensitivity)(prevalence)
(sensitivity)(prevalence) + (1-specificity)(1-prevalence)

According to your data:
prevalence = 0.1 [100 cases of cancer in 10,000 persons]
sensitivity = .99 [99% of cancer cases will test positive]
specificity = 0.1 [1% of people will test false positive]

Plugging these numbers in the formula gives a quotient of 0.5, or a 50% PPV.

 

[andross77]

wow, you guys are smart. my math skill experienced a horrible, agonizing death in high school calculus...

 

[lawtonfogle]

50%!!!
Either he has, or has not got cancer!


Serious though I would assume: 100 - (1 false positive) * 99% = 98.01.

Although I'm not sure if you mean that 1% false positive is in- or excluded in the 99% correctness.
It depends on: "If you actually do have cancer". Do you mean actually or actually according to the (incorrect) test?

It is excluded. That is, if you do have cancer, you work with the 99%, if you don't have cancer, you work with the 1%. So they are separate.

Good job.

 

[lawtonfogle]

wow, you guys are smart. my math skill experienced a horrible, agonizing death in high school calculus...

I suggest you give calculus another chance, just make sure you get a great teacher.

 

[Ectezus]

It is excluded. That is, if you do have cancer, you work with the 99%, if you don't have cancer, you work with the 1%. So they are separate.

Good job.

Thank you but now I'm even more confused. Which I guess is the whole point of your topic.
Just for clarification: So you're talking about the actual truth, not the test results when you talk about "When you do have cancer"? This changes the whole thing really.

If the false positive rate is 1% then that already means out of 10.000 a 100 people will test falsy positive, but we don't know how much of that 1% applies to that 10.000 group because we don't know how much actually have cancer in the first place.

Basically in essence, we also need to know the normal occurance of cancer before we can apply the numbers correctly. Similar to the previous Bob example where we need to know the normal occurance of females and males that are named Bob.


Is there a particular point you would like to make by the way?
Call me paranoid but I suspect something like "We can't make 100% accurate predictions without knowing the absolute truth which is almost impossible to know, therefore science can be wrong". (Probably followed by: therefore god exists )

- Ectezus

 

[Stormy]

Well, you guys so far did great on the first one, so here is the second one.

We know that 100 out of the 10,000 people tested have cancer.

We also know that the test, if you actually do have cancer, is 99% correct.
We also know that the test, if you don't have cancer, will give a false positive 1% of the time.

Bob (same Bob from before), just received a positive result on this test. What is the chance that Bob has cancer?

Bob should be re-tested. At this time there is a mere 1% chance that the test was wrong. There is a 99% chance that the test was accurate and Bob does have cancer.


What does this have to do with the morality forum?

 

[Mystman]

Well, you guys so far did great on the first one, so here is the second one.

We know that 100 out of the 10,000 people tested have cancer.

We also know that the test, if you actually do have cancer, is 99% correct.
We also know that the test, if you don't have cancer, will give a false positive 1% of the time.

Bob (same Bob from before), just received a positive result on this test. What is the chance that Bob has cancer?

There are 9900 people without cancer. 1% false positives = 99 people
There are 100 people with cancer. 99% will get that result = 99 people

So yeah, 50% chance that Bob actually has cancer.

 

[Ectezus]

There are 9900 people without cancer. 1% false positives = 99 people
There are 100 people with cancer. 99% will get that result = 99 people

So yeah, 50% chance that Bob actually has cancer.

But if 9900 of the people have no cancer then 99 of those 9900 will show positive, which still leaves 100 people (with actual cancer) to complete the 10.000 which would make the grand total a 199. (While we know its only 100)

Basically, if 10.000 of the tested people actually have no cancer. It would still show up as 100 due to the 1% false positive.

Maybe the answer that bob has cancer is 0% because it's the only possible way for the 10.000 number to not contradict the the 1% false positives statement...

Aaagh, stupid question.

- Ectezus

 

[Mystman]

But if 9900 of the people have no cancer then 99 of those 9900 will show positive, which still leaves 100 people (with actual cancer) to complete the 10.000 which would make the grand total a 199. (While we know its only 100)

Basically, if 10.000 of the tested people actually have no cancer. It would still show up as 100 due to the 1% false positive.

...

What?

Of every 100 sick people, this test wil mark 99 as sick.

Of every 100 healthy people, this test will mark 1 as sick.

In this situation: 10.000 people.
100 of them sick -->> test will mark 99 as sick
9900 healthy -->> test will mark 99 as sick

end-result: 50% of the population that is marked as sick is actually sick, and 50% is not.

So Bob has a 50% chance of being sick.

(I honestly have no idea what you were trying to say in your post )

 

[lawtonfogle]

Bob should be re-tested. At this time there is a mere 1% chance that the test was wrong. There is a 99% chance that the test was accurate and Bob does have cancer.


What does this have to do with the morality forum?

How people reason has is crucial to dealing with how they think though ethical or moral problems. This is a window into how people reason, thus a window in how they deal with such problems.

 

[Ectezus]

In this situation: 10.000 people.
100 of them sick -->> test will mark 99 as sick
9900 healthy -->> test will mark 99 as sick

My point was that 99+99 is 198 and I thought the OP said the test results showed 100 people have cancer. In which case the numbers don't add up.

We know that 100 out of the 10,000 people tested have cancer.

But now that I read the sentence again I'm not so sure anymore. Basically I read "tested to have cancer" in which case the test produces 100 positive results instead of "100 really have cancer no matter what the test says".

Lawtonfogle, Do you mean those 100 people really have cancer or the test shows they have cancer?
Ugh I hate misinterpretations.
If it's 100 as an absolute truth then yes, its clearly 50%.

- Ectezus

 

[lawtonfogle]

My point was that 99+99 is 198 and I thought the OP said the test results showed 100 people have cancer. In which case the numbers don't add up.



But now that I read the sentence again I'm not so sure anymore. Basically I read "tested to have cancer" in which case the test produces 100 positive results instead of "100 really have cancer no matter what the test says".

Lawtonfogle, Do you mean those 100 people really have cancer or the test shows they have cancer?
Ugh I hate misinterpretations.
If it's 100 as an absolute truth then yes, its clearly 50%.

- Ectezus

The scientist know this is the 100_per_10,000 cancer. They know 100 of every 10,000 will get it, this is not based on the test but on other knowledge.

 

[Ectezus]

The scientist know this is the 100_per_10,000 cancer. They know 100 of every 10,000 will get it, this is not based on the test but on other knowledge.

Thanks for clearing up it's not the from the actual test results itself.
The 100 out of every 10,000 only makes it an average though. (Instead of a certainty) There can still be tests where the number of people with cancer is twice as much or twice as few.

I'm not sure if those 100/10,000 odds are taking from the whole world or just America. But in any case, if our hypothetical Bob lives near Chernobyl I would say his odds are much larger than 50%.

- Ectezus

 

[Stormy]

How people reason has is crucial to dealing with how they think though ethical or moral problems. This is a window into how people reason, thus a window in how they deal with such problems.

On a purely mathematical basis there is a 11% chance that Bob has cancer.

I see the correct answer as a stupid answer that may allow people to not take the cancer test seriously. I would rather Bob think that there is only 1% chance that the test is wrong and then be happy if it indeed is a false positive. Sometimes being smart can make people behave stupidly.

 

[Mystman]

On a purely mathematical basis there is a 11% chance that Bob has cancer.

I see the correct answer as a stupid answer that may allow people to not take the cancer test seriously. I would rather Bob think that there is only 1% chance that the test is wrong and then be happy if it indeed is a false positive. Sometimes being smart can make people behave stupidly.

(where does the 11% come from?)

Anyway, the 50% certainly isn't "stupid". Taking this kind of stuff into account is very important, both for doctors and patients.

These days, you have lots of people undergoing lots of tests, sometimes even when there are no signs of disease. Being declared sick, when you're actually quite healthy, leads to a number of bad things:

1. Stress in the patient (I'd prefer "There's a 50% chance you're sick" over "You're going to die!11!!")
2. Doctors spending time/money on a healthy person, time/money that could've been spent on sick persons, thus leading to less survival chances of the actual sick persons.
3. The patient sometimes actually undergoes treatment, where the treatment's side-effects may be harmful. (say... a test indicated that person X has something wrong with his aorta. A minor operation is done to check what exactly is wrong, and possible correct the problem. It turns out there is no problem. And then the wound gets infected and the patient dies. It happens.)

And probably some more. Given that in any population, the number of healthy people is often much higher than the number of sick people, these effects might lead to preventive testing actually doing more harm than good.

 

[Ectezus]

I see the correct answer as a stupid answer that may allow people to not take the cancer test seriously. I would rather Bob think that there is only 1% chance that the test is wrong

Lets not forget there is a difference in "Whats Bobs chance to actually have cancer ONCE he's diagnosed" and "Whats the chance for Bob to receive a positive test result?"

The chance (on average) to have cancer or not once you've already tested positive is 50%.

However:
The chance to be tested positive is only 1.98%!

(100 have cancer -> 99% will be picked out of the test.
9,900 healthy -> 1% chance to get positive test result.
Soo 99 + 99 = 198/10,000 = 1.98%)

So of course when you take a cancer test the chance is not 50-50... The test does mean something because of the small chance to be tested positive in the first place!

If you combine those numbers together it will be a 0.99% chance to tested positive and not have cancer, and a 0.99% chance to be tested positive and do have cancer.
(And a 0.1% chance to be tested 'cancer free' while you actually do have cancer. Of course, doing two tests practically eliminates all problems.)

I would rather Bob think that there is only 1% chance that the test is wrong

Well there you go, you don't have to 'rather think', now you know. It's actually less than 1%.

- Ectezus

 

[Stormy]

(where does the 11% come from?)

I did make a mistake, the correct answer is actually 2%. I was using 10% instead of 1% for false positive. oops

Not able to draw a graph on this post to make the work easy to see. Hope you can follow this...


1. We know that 100 out of the 10,000 people tested have cancer.

100 have cancer ... 9,900 do not have cancer

2.We also know that the test, if you actually do have cancer, is 99% correct.

Of the 100 who have cancer.. 99 have correct positive test

3.We also know that the test, if you don't have cancer, will give a false positive 1% of the time.

Of the 9,900 who do not have cancer 99 will have positive test.


Total positive test received 99 + 99 = 198
Total positive test actually cancer = 99


99 is 2% of 198

In this sample a positive test has an accuracy of 2% correction

 

[Mystman]

3.We also know that the test, if you don't have cancer, will give a false positive 1% of the time.

Of the 9,900 who do not have cancer 990 will have positive test.

1% of 9900 is 99, not 990.