### Doctor House, Please Meet Mister Bayes

Let’s assume there’s some deadly disease out there, but there’s a pretty good test for it which is 90% accurate (if a person is sick – the test will claim he’s sick in 90% of the cases, and if a person is healthy – the test will claim he’s healthy in 90% of the cases). Now, assuming you took this 90% accurate test, and got a positive result implying you are sick - what are the odds that you are really sick?

Think of your answer before proceeding.

Most of you are probably thinking about 90%, and obviously, that's wrong (otherwise this post would be pointless). I intentionally left out a crucial piece of data which is that this disease affects 1 in every 10 people. Like many problems such as this, it looks very simple, but since most people never heard of Bayes’ theorem, they will only consider the test’s accuracy, ignoring the prior likelihood of the disease. I could use the formula, but I think using an example is much simpler. Let’s say 100 people take this test. Since the disease affects 10% of the population, 10 are really sick. The accuracy of the test is 90%, so 9 of them will get a positive result (true positives). From the other 90 people which are healthy, 81 will get a correct negative result, but 9 will get a positive result although they’re actually healthy (false positives). So after testing 100 people you end up with 18 people with positive results, but only 9 of them are true positives. 9/18=0.5=50%, so, the final answer is: Assuming you took this 90% accurate test to a disease that affects 10% of the population, and got a positive result implying you are sick – the actual odds that you are really sick are only 50%. Several studies where similar problems were presented to real doctors, demonstrated that only about 15% of them get it right. Isn't that amazing? 85% of the doctors will tell you you've got a pessimistic 90% chance of dying really soon, when actually there's a semi-optimistic 50% chance that there's nothing wrong with you! Now that you know this, will you ever be able to trust your doctor again when probabilities are involved?

~~~

p.s. I think, therefore I am wrong.

Think of your answer before proceeding.

Most of you are probably thinking about 90%, and obviously, that's wrong (otherwise this post would be pointless). I intentionally left out a crucial piece of data which is that this disease affects 1 in every 10 people. Like many problems such as this, it looks very simple, but since most people never heard of Bayes’ theorem, they will only consider the test’s accuracy, ignoring the prior likelihood of the disease. I could use the formula, but I think using an example is much simpler. Let’s say 100 people take this test. Since the disease affects 10% of the population, 10 are really sick. The accuracy of the test is 90%, so 9 of them will get a positive result (true positives). From the other 90 people which are healthy, 81 will get a correct negative result, but 9 will get a positive result although they’re actually healthy (false positives). So after testing 100 people you end up with 18 people with positive results, but only 9 of them are true positives. 9/18=0.5=50%, so, the final answer is: Assuming you took this 90% accurate test to a disease that affects 10% of the population, and got a positive result implying you are sick – the actual odds that you are really sick are only 50%. Several studies where similar problems were presented to real doctors, demonstrated that only about 15% of them get it right. Isn't that amazing? 85% of the doctors will tell you you've got a pessimistic 90% chance of dying really soon, when actually there's a semi-optimistic 50% chance that there's nothing wrong with you! Now that you know this, will you ever be able to trust your doctor again when probabilities are involved?

~~~

p.s. I think, therefore I am wrong.

## 13 comments:

That sounds really cool !! there was a famous case in about death of two kids.

http://www.timesonline.co.uk/tol/comment/columnists/minette_marrin/article1530586.ece

There's also a striking, fictitious example here:

http://www.sffaudio.com/?p=2710

interesting concept. I think it goes with the saying "you can see the cup as half full or half empty."

sidd.

I think it's crazy however, sadly probable.

Great Blog..

Dorothy from grammology

grammology.com

The thing is that there are usually other indicators than a test. If your doctor is relying elusively on the result of a single test you need to fire him or her. But this is interesting to think about.

Actualy doctors should need to diagnose with both experience & test result.....

Iam very bad in math..but the writing was cool.

:) nice post

GOD I'M LOVING YOUR MIND!!! :D :D

I should show this to my mother. :D

I believe that you should still trust your doctor. After all, it is better to be safe than not prepared...

Just found your blog on blogcatalog and I like what I see!!! Keep it up!

Philosophical Blog of a Modern Teenager

Nice instructive post.

simple example for an important and confusing subject!

will go now to read some other posts.

Gilad

I enjoyed this post Uri. As you rightly assume, most people have only a sketchy understanding of probability and will reach conclusions that are inherently

improbable.I don't know if you've encountered the three boxes problem before, but if you haven't, you might like to check out my post on probability.

Nice one. I thought it was gonna be something like 80%. Anyway I learnt something new. the baye's theorem. Thanks for the heads up.

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