Cancer probability

[Juked07]

Originally posted by: Numenorean

Originally posted by: Leafy

Originally posted by: Numenorean

If you receive a positive result, then 98% of the time you will have cancer. Both ways you asked the question in the OP are asking that chance based on you receiving a positive test result.

I asked what chance you have of >>actually<< having cancer if your test is positive.

Originally posted by: Numenorean
You didn't say anything about having cancer, not that it matters.

Originally posted by: Leafy
(No, this isn't a homework problem. Just for the lulz of everyone who will get it wrong)

If 0.5% of the population have cancer, and there is a 98% accurate test (98% who have cancer test positive), and you receive a positive result, what is the probability of your having cancer? (at least 4 decimal places)

25 cents to whoever solves it first.

Phrased differently, What is the probability you have cancer given a positive test result?

Originally posted by: Numenorean
If you have cancer and are given a positive test, it's still 98% - if you KNOW you have cancer then it's 100% because you already know and then the test won't matter. If you are given a false result and you KNOW you have cancer you still have a 100% chance of having cancer.

You can't look at your body and say "Hey, I see lung cancer!". YOU don't know anything of the state of your having cancer until you take the test.

Originally posted by: Numenorean
Now if you are going to ask a DIFFEFERENT question - i.e. If you KNOW you have cancer and are tested, what is the chance that the test result comes back positive? That's a completely different question. But it still has the same answer. 2% of the time, the test will fail. In this case, return a false negative (since you know you have cancer the negative is false) - so still you have a 98% chance of receiving a positive result, even though you already know you have it. Knowing you have it doesn't change the accuracy of the test.

OP = FAIL

You = fail hard. How can you miss this every time I phrase it?

And again you're changing your mind??

I asked what chance you have of >>actually<< having cancer if your test is positive.

Okay...this has already been answered. You said the test is 98% accurate. If your test comes back as positive, then you have a 98% chance of having cancer. If that's incorrect, then the test isn't 98% accurate as you state.

Your response to my statement that you didn't say anything about having cancer is just you highlighting my point. You were asking about your chances of having cancer, not saying that you do have cancer which is what I was pointing out because later in the post you said this:

Fail. If the question was "What is the probability that you test positive given that you have cancer"

So now you are asking the probability of a positive test result if you DO have cancer - not what the probability is that you have cancer if you do test positive. Can you see the difference?

You = FAIL

lol, you're so wrong, and yet so confident you're right.

Really brilliant work.

Edit: Glancing at most of the other posts, ATOT fails hard in this thread in general >.>

The OP's posts are entirely correct given that you successfully interpret "98% accuracy rate" as 98% accurate whether the result is positive or negative, which is an obvious assumption to make. Pointing to that phrasing as ambiguous is stingy nitpicking at best. It's a simple Bayes' Rule problem, and 99% of ATOT can't solve it. =(

 

[IGBT]

..it's out there waiting for all of us. all you have to do is live long enough or expose your self long enough.

 

[BrownTown]

Originally posted by: Numenorean
Okay...this has already been answered. You said the test is 98% accurate. If your test comes back as positive, then you have a 98% chance of having cancer. If that's incorrect, then the test isn't 98% accurate as you state.

Main, I know I'm being an asshole here, and I do apologize for that, but you are just failing all over yourself here. Just read Leafy's explanation, its spells it out so that anyone should be able to follow. What you are apparently missing is that the 99.5% of well people all ahve a 2% chance of receiving a positive test result. Since this is a much LARGER group that means that the majority of positive tests are false positives from the group of people who are not sick. You just have to realize that .995 * .02 (chance of false positive) is much larger than .98 * .005 (chance of real positive). Pretty much you are just turning yourself into that guy from the airplane on a treadmill thread who kept it going 1000 posts longer than it needed to.

 

[Juked07]

Originally posted by: BrownTown

Originally posted by: Numenorean
Okay...this has already been answered. You said the test is 98% accurate. If your test comes back as positive, then you have a 98% chance of having cancer. If that's incorrect, then the test isn't 98% accurate as you state.

Main, I know I'm being an asshole here, and I do apologize for that, but you are just failing all over yourself here. Just read Leafy's explanation, its spells it out so that anyone should be able to follow. What you are apparently missing is that the 99.5% of well people all ahve a 2% chance of receiving a positive test result. Since this is a much LARGER group that means that the majority of positive tests are false positives from the group of people who are not sick. You just have to realize that .995 * .02 (chance of false positive) is much larger than .98 * .005 (chance of real positive). Pretty much you are just turning yourself into that guy from the airplane on a treadmill thread who kept it going 1000 posts longer than it needed to.

Legit.

 

[RapidSnail]

Originally posted by: BrownTown
Its amazes me ever time that on a board that tries to act so educated so many people can get such a simple probability question wrong...

Not to mention the fact that its a classic problem many people should probably have heard before.

I read your responses and it makes sense. The probability of one person in a population having cancer is small, and the probability of false positive is higher.

However, I've never heard of Bayes' rule or the problem, nor was I able to solve it mathematically. Logically it makes sense.

 

[dullard]

Originally posted by: Leafy
(No, this isn't a homework problem. Just for the lulz of everyone who will get it wrong)

If 0.5% of the population have cancer, and there is a 98% accurate test (98% who have cancer test positive), and you receive a positive result, what is the probability of your having cancer? (at least 4 decimal places)

25 cents to whoever solves it first.

Phrased differently, What is the probability you have cancer given a positive test result?

Since you never gave the full statistics, this question is unanswerable.

Later in the thread you state that the accuracy for people who don't have cancer is 98%, but from the OP, there is no reason to assume that to be true. Leafy, you can't just make up data later.

There are 4 possibilities, since cancer is a binary condition
1. Test shows you're positive, you're positive (Accurate, 98% chance)
2. Test shows you're positive, you're negative (inaccurate, 2%)
3. Test shows you're negative, you're positive. (inaccurate, unknown %)
4. Test shows you're negative, you're negative (Accurate unknown %)

Percentages #3 and #4 were never given in the OP and in real life it is HIGHLY UNLIKELY that they will be the same as #1 and #2.

Suppose a "test" just randomly said 98% of people who took the test were cancer positive. Then, it would be true that 98% of people who have cancer will test positive. However, 98% of people who DON'T have cancer will test positive in this case. Using your 10000 people example:
1. 50 people have cancer. 49 will likely test positive (accurate 98%).
2. 9950 people don't have cancer. 9751 will likely test postitive (inaccurate 98%).
3. 50 people have cancer. 1 will likely test negative (inaccurate 2%).
4. 9950 people don't have cancer. 199 will likely test negative (accurate 2%).
Thus, 49 actual cancers in the postive group out of 9800 people tested positve = 0.5000% as the answer to your question in this "test".

 

[ivan2]

marked for ownage thread of the year 2009 candidate.

 

[yelo333]

Originally posted by: ICRS

Originally posted by: yelo333
Text

Thanks for the quick bayes' rule refresher, Leafy!

Yes, the wording about 98% positive threw me at first, but I knew what you meant. (as do the others...)

There is a problem in your solution you assumed something which we don't know to be true.

P(T|C') != 1 - P(T|C) .

Yes, the following sentence is missing in the OP: Also, the test will report a negative result for those without cancer 98% of the time.

I assumed it because I treated it like a test question a forgetful professor would write. If the professor was more tricky than forgetful, it would be better to go the not enough information route.

So, it would probably be best to say something like:

As the question is stated, there is not enough information to solve the problem. If it is further assumed that the test will report a negative result for those without cancer 98% of the time, then...

[insert solution]

 

[yelo333]

Originally posted by: ivan2
marked for ownage thread of the year 2009 candidate.

...for forgetting one sentence?

 

[ivan2]

Originally posted by: yelo333

Originally posted by: ivan2
marked for ownage thread of the year 2009 candidate.

...for forgetting one sentence?

still see no reference to false negative numbers. 98% accuracy to cancer patient doesn't imply a 98% accuracy to non-cancer patients.

ok we are over thinking a high school level problem, fail to the rest of us.

 

[Mark R]

Not enough info.

You have quote the specificity of the test as 98% (incorrectly calling this accuracy). The sensitivity of the test is also required before the positive predictive value can be calculated.

Or are you asking us to assume a sensitivity of 98%? (On the basis that specificity and accuracy are both 98%, and therefore sensitivity is too)

If you assume the above, then the positive predictive value, in the specified population (i.e. the probability of having cancer given a positive test) is 19.75%. In other words, 80.25% of positives are false.

 

[Mo0o]

Originally posted by: yelo333

Originally posted by: ivan2
marked for ownage thread of the year 2009 candidate.

...for forgetting one sentence?

It's a pretty big sentence to forget and simply assume people will know what he means. This coupled by the fact that he was really arrogant about it makes it a candidate

 

[Oyeve]

100%. If you live to a nice old age and die basically of old age I bet you have a cancer of some sort in your body.

 

[A Casual Fitz]

Originally posted by: BrownTown

Originally posted by: Numenorean
Okay...this has already been answered. You said the test is 98% accurate. If your test comes back as positive, then you have a 98% chance of having cancer. If that's incorrect, then the test isn't 98% accurate as you state.

Main, I know I'm being an asshole here, and I do apologize for that, but you are just failing all over yourself here. Just read Leafy's explanation, its spells it out so that anyone should be able to follow. What you are apparently missing is that the 99.5% of well people all ahve a 2% chance of receiving a positive test result. Since this is a much LARGER group that means that the majority of positive tests are false positives from the group of people who are not sick. You just have to realize that .995 * .02 (chance of false positive) is much larger than .98 * .005 (chance of real positive). Pretty much you are just turning yourself into that guy from the airplane on a treadmill thread who kept it going 1000 posts longer than it needed to.

Damn, I gotcha now. D'Oh

 

[Q]

The answer is waffles