This is another puzzle from the Ramanujan collection, a nice probability puzzle!
The World Health Organization has just identified a new disease. The good news is that it is very rare: only 1 in every million people has it. The bad news is that it is very deadly.
There is a test available that can tell you whether or not you have the disease. It gives the correct diagnosis 99% of the time. This means that if someone with the disease is tested, the test will say “positive” 99% of the time and “negative” 1% of the time; if someone without the disease is tested, it will say “negative” 99% of the time and “positive” 1% of the time.
You take the test, and it comes up “positive”. Oh no! Before you start to panic, what is the probability that you definitely have the disease? Round your calculation to the nearest whole percent.
As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.
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P( diseased | result is positive ) = P ( diseased intersection result is positive ) / P(result is positive)
= 1/ 10^6 * (0.99) / ((1/ 10^6 * (0.99)) + ((1-1/10^6)*0.01))
= 0.99 / (0.99 + 0.01*(10^6-1))
= 0.99 / (0.98 + 10^4)
~10^(-4) ~ 0.01%
0.01%