Probability theory and your last medical test result

Here is a probability problem based on http://en.wikipedia.org/wiki/Bayes'_theorem

Suppose you were tested for colon cancer and the test results are positive. Then what is the probability of your actually having colon cancer. You asked for data about accuracy of the test and here are they:

1. The test gives 10% false positive. i.e. Take a population of 100 healthy people, still the test will tell 10 have cancer.
2. The test gives 1% false negative i.e. Take a population of 100 confirmed colon cancer cases, still the test will tell 1 has no cancer.
3. The population to which you belong to has .1% probability of colon cancer i.e. 1 in 1000 has colon cancer.

The simple answer in your mind says that you have 90% chance of having cancer, because 10% false positive, i.e. 90% true positive. But that is not the case. To understand:

1. Let's say you take a population of 100000.
2. Then with 0.1% probability 100 had cancer and the test detected 99 of them as having cancer.
3. Also the test detected falsely 10% of all healthy i.e. 10% of 99900 as having cancer, i.e. 9990 as having cancer falsely.
4. So total population of cancer +ve by this test is 9990 + 99 = 10089 out of which 99 actually have cancer 
5. Now you belong to this group of having cancer and so your probability is 99 / 10089 i.e. 0.98%age i.e. roughly 1 in 100 chance.

Now what do you do after cancer detection. The Doctor says go for a 2nd test to make sure. Now consider the 2nd test is independent of the first one. Then if it detects +ve again then the joint probability of both detecting false cancer is, .99 * .99 = 0.98 i.e. chance improves to 2 in 100. Will you do 100 tests now! Also consider the 2nd test may not be totally uncorrelated to the first test. Then it will not even confirm to 2 in 100.

So next time a test result comes +ve, don't panic. The Doctor is not a statistician. Cool down and try to find a way to confirm either way.