Please note - this is just a puzzle. The ILLUSIONS virus does not really exist!
Government scientists have established that a very serious virus, the ILLUSIONS virus, has infected the population. At present it shows no symptoms, and it is not known how it spreads from person to person. The government scientists, using very sophisticated equipment, have established for certain that 1 person in a thousand is infected.
Your local chemist is offering a do-it-yourself testing kit, which is simple and cheap. However it is only 95% accurate, i.e. if you have the virus there is a 95% chance it will predict this correctly, and a 5% chance it will be wrong, and tell you you do not have the virus. Likewise, if you do not have the virus, there is a 95% chance it will tell you that you are OK, and a 5% chance it will tell you that you have the virus.
You buy a kit, and it tests positive, i.e. it is telling you that you have the virus.
What is the actual probability that you have the ILLUSIONS virus?
The answer may surprise you - the actual probability that you have the virus is less than 2%!
How does this come about?
Well, basically the virus is quite rare - only 1 person in a thousand actually has it. On the other hand, since the test is not very accurate (95% accuracy is really not all that accurate) there will be many more 'false positives' - people who do not really have the virus, but who are the 5% who test postive - so that the false positives swamp the real positives.
Let us do the maths.
Take a random 20,000 people from our population. (I have not specified what country this - mythical - virus has struck, so we do not know the actual population size. The maths works with any number, I have just picked a figure that makes it easy!)
Of our 20,000 population, 20 will actually have the virus (1 in a thousand have the virus). If these 20 all take the simple test, 95% will be correctly diagnosed as infectious, which is 19 people, and 1 person (5%) will come up as a false negative, i.e. the test will say they do not have the virus, even though they do.
Of the remaining 19,980 members of the population who do not have the virus, 95% will test negative, that is 18,981. 5% will test as positive, i.e. having the virus, even though they do not. That will be 999 people.
So, there will be 19 people who have tested positive who really do have the virus, and 999 who have tested positive who do not have the virus.
So there will be a total of 1018 people who have tested positive for the virus, of whom only 19 actually have the virus. 19 divided by 1018 is 1.87%.
I was recently introduced to one of the leading malaria experts in the UK, and he got the puzzle immediately. He also said that this has an exact parallel in the real world, when you have a disease that is quite rare and the test is not 100% accurate - the false positives tend to swamp the genuine positives.
Photo courtesy of kreg.steppe on flickr.>/p>