If the mod thinks that this might belong to the homework thread, please be free to move it. I just thought that it would generate some great discussion... And not a homework problem anyway =)

The following is from Derren Brown's, Tricks of the Mind; Thinking Traps. and if you like this little piece; please go and get the book as it's pretty fascinating.

“Imagine there is a terrible disease reported, and although it affects only one in ten thousand people, it is absolutely lethal. You are worried about it, so you decide to undergo a medical test to see if you have the disease. Now, no medical test is ever 100% accurate, but your doctor explains that this one is known to be 99% accurate, regardless of whether or not you have the disease (in other words it will deliver a correct positive or negative result 99% of the time).

You decide to take the test. You are a little nervous, but you think it’s a sensible thing to do. A blood sample is taken and you’re told the results will be sent in the post.

A week later the envelope arrives from the testing centre, you open it up… staring you in the face is the answer you dreaded, the result is positive. The test has indicated that you have the lethal disease... How likely are you to have the disease?

Answer:

"You are less than 1% likely to have the disease.
The disease hits only 1 in 10,000 people. So your positive test result, with its 99% accuracy, could mean that you are one of the 99% of people who have been correctly told they have the disease, or you could be one of the 1% of people who don't have the disease but have been wrongly told that they do. (Remember the test is 99% accurate, regardless of whether or not you have the disease.) So, are you more likely to be one of the correctly diagnosed people with the disease, or one of the wrongly diagnosed people without it?

The disease strikes only one in 10,000. So, forgetting that for a moment, you can immediately see that you are far more likely (to the tune of 9,999:1) not to have the disease. Now, imagine that a million people take the test. Only a hundred or so of those will actually have the disease. 99 out of those 100 will be correctly diagnosed as having it because the test is 99% accurate. On the other hand, 999,900 people won't have the disease, but 1% (or 9,999) of them will be wrongly diagnosed as having it. So, are you one of those 99 who have it, or one of those 9,999 who don't You're over a hundred times more likely to be in the second, safe, category."

I keep on thinking that the answer is 1%, because that is the percentage that the test will give you a false positive.

Now these are some of the things I don't understand;

In a way I understand what Derren is trying to say; Before you go to the test you have 1 in 10,000 chance of having the disease so it really shouldn't change the absolute truth of whether you have the disease or not, just by going to take a test. But I don't think that this is the key to solving the problem.

And...

Why does the first statistical number weight more value than the test it self?

My counter argument would be; lets say your wife was pregnant and there's a 50/50 chance that it's either a boy or a girl... Now you go to the doctors to get a test and the test is 99% accurate. The test comes back saying that it's a girl... I doubt anyone would still come up to me and tell me that the chances of it being a boy or a girl is still 50/50...?

Cliffs:
- There's a lethal disease that kills 1 in 10,000 people
- There is a test that is 99% accurate to whether you have this disease or not
- You take the test and it comes back positive
- How likely are you to have this disease?

Any input/argument/debate is much appreciated.

Thanks!

Consider your boy/girl example for a moment. Take a thousand pregnant women, all of whom take the test:

495 women will have a boy and the test will say boy
005 women will have a boy and the test will say girl
495 women will have a girl and the test will say girl
005 women will have a girl and the test will say boy.

Suppose you're a women whose test says boy. You're either one of the 495 women who's having a boy, or one of the 5 women who's having a girl.

Now consider the original example:
99 people will be positive and the test will say positive
1 person will be positive and the test will say negative
9999 people will be negative and the test will say positive
989901 people will be negative and the test will say negative.

Suppose your test says positive. You're either one of the 99 people who's positive, or one of the 9999 people who's negative.

See the difference (assuming my math's correct)?

http://en.wikipedia.org/wiki/Bayes'_theorem

This.

Either you have the disease or you don't, so the probability must be 1/102.

50/50 IMO.

Bayes' theorem is easiest from an odds perspective IMO. 2 possibilties:

a)You have the disease and the test was right. This has probability
1/10000 * 0.99 = 0.0099%

b) You don't have the disease and the test was wrong. This has probability
9999/10000 * 0.01 = 0.9999%

So the odds against having the disease are

0.9999:0.0099 = 0.0099 / (0.0099 + 0.9999) = 0.98%

Using the term "accurate" to describe the test is surprisingly imprecise. Is that a 99% sensitivity? specificity? Or a positive predictive value? Or a likelihood ratio?

I am a math fish

If you write this you don't understand what's going on.

Conditional on testing positive, you either have the disease or you don't. It turns out that because having the disease is much rarer than getting a false positive on your test, it is much more likely that you got a false positive (see Bruce's post for the exact calculations). If, on the other hand, the test is accurate 99.9999% of the time, then false positives are much rarer than 'true' positives, and you are likely to actually have the disease.

Watch this video:

http://www.ted.com/talks/peter_donne...ol_juries.html

This speaker explains the question in the OP, and a few other little things. And he is a good speaker. Enjoy.

And in case anyone's concerned about the results of their own medical tests, this is one of the reasons we don't do every test on every person. The tests become useful when they're done to populations who are more likely than average to have the disease.

Based on his later explanation, it looks like 99% positive predictive value and 99% negative predictive value.

The other comments cover what you've gotten wrong technically, but one line from your post in particular stood out to me as to what you're misunderstanding

You seem to be thinking like we just ignore the test and just look at the original probability, but that doesnt happen at all. Sure, after you take the test there's still less than 1% chance you have the disease, but thats up from the 0.01% chance you had before. The positive result has raised the chance you have it nearly 100 fold.

The reason it seems like the test did not do much is because once you have a the result, you know that one of 2 'rare' things have happened. Either you're one of the .01% who have the disease, or you're one of 1% who dont but get misdiagnosed. Clearly one of these is still much rarer.

Lastly, consider what happens if you take the test again and get given positive again (assuming 1st and 2nd test chances of being wrong are independant). Now the chance you have the disease is 49.5% by my math, which is about right because the 2 'rare' events that could have happened are both 0.01%.

In this thought experiment, it's safe to assume that we have access to God's perspective and know exactly how good the test is. We don't need to resort to statistical statements that involve quantities that mortals can measure.

I took 99% in this context to mean:

-If you have the disease, God rolls a 100 sided dice. If he gets 1-99, the test issues a positive, if he gets 100, it issues a negative

-If you don't have the disease, God rolls a 100 sided dice. If he gets 1-99, the test issues a negative, if he gets 100 the test issues a positive.

Damn... Thanks a lot guys =)

I think I'm finally starting to get it!

After reading the problem, I got this feeling as if the test results were ignored and that was a bit unsettling.

But thanks again for all the explanation and help! =) Made my nights a lot more peaceful!

Ooh! =) Thanks! I love TED Talks!!

Think of the test results as "updating" information that you had previously. Before you took the test, you knew there was some very small probability that you had the disease. After you took the test, that probability was increased, but not to a large amount since your initial probability was so low.

On the other hand, if you're showing signs of having a certain disease, your probability of having that disease is going to be higher than the average person. If you then take a test and get a positive result, in the end you'd be very likely to have the disease since your previous chances of having the disease (based on your symptoms) was quite high.

I take this back. Will re-work this later.

Really enjoyed this video. Thanks!

I hate the thing he did with the coin flipping. The whole way through the show, in the back of my mind, I was like, "how ridiculously stupid would it be if he had done this experiment 10's of 1000's of times?". And then 5 minutes later...BOOM! Derren Brown ruined my day because his stuff is usually good fun.

Fm surprising results. Think I'm going about this in the right way....

The test can have 4 outcomes:

1.Negative and Test Correct
2.Negative and Test Incorrect
3.Positive and Test Incorrect
4.Positive and Test Correct

For the odds that both situations will occur simultaneously you have to multipy them, so:

1. Neg x correct = 0.9999 x 0.9899 = 0.98980101 = 99%
2. Neg x Incorrect = 0.9999 x 0.01 = 0.009999 = 0.99%
3. Pos x Incorrect = 0.0001 x 0.01 = 0.000001 = 0.0001%
4. Pos x correct = 0.0001 x 0.99 = 0.000099 = 0.0099%

This is the best way to explain it imo.

So how did Derren Brown do his Assasin trick last night?

http://www.zimbio.com/Hypnosis/artic...+Explained+How

This article explains what happened but doesn't give an answer. Has anyone got any idea? It can't be a hoax (as that would be stupid) and I can't see a way it can be in between real and fake...

Here's a non-mathy explanation that will hopefully help you get some sleep--

Suppose we have a test that tests if someone takes a drug. If someone took the drug, the test will say they did 99% of the time. If someone didn't take the drug, the test will say they didn't 99% of the time.

You take the test and the test indicates that you have taken the drug.

Now, imagine the drug is 'rocks from jupiter'. Since no one has ever eaten a rock from jupiter, the probability that you actually took this drug is 0%.

Now, imagine the drug is 'water'. Since everyone needs water to survive, the probability that you actually took this drug is 100%.

So, the answer to this type of question can range anywhere between 0% and 100%, depending on the percent of the population that actually takes the drug.

^^ very nice

Keep thinking about it until you do understand it, grokking this point will help you in so many other areas.

It seems very unnatural at first, it seems our brains end up wired to think in a certain way that's perhaps computationally easier, but at odds with reality in many cases; until we realize the error and consciously re-train.

A summary of what's going on here:
- You think the probability is {some figure}
- You get some more information
- You now think the probability is {some figure modified to take into account the new information}.

The key point is that the new value depends on what you thought it was before doing the test. The test has refined your knowledge.

Your original suggestion that the test result implies that you have a 99% chance of having the disease, would actually mean that you are not using the original piece of information at all in reaching your answer, so this should clue you in that you aren't applying the right logic.

This actually happens in real life with Aids tests, there's a lot more false positives than real positives, so the first test shouldn't send you into a panic.