A Probability Puzzle

The Puzzle

I discovered an extremely interesting probability puzzle a couple days back which is very relevant to the past year. The puzzle is known as the “Lumps have hit Flatland” and is featured on Brilliant. The contraints are this:

1. At any given time, 1/5 of the people in Flatland have the Lumps.
2. When someone has the Lumps, the test always comes back positive.
3. When someone does not have the Lumps, the test comes back positive 25% of the time.

So, if you recieve a positive test, what is the chance you have the Lumps?

Solving The Puzzle

Let’s assume that you get a positive test and you are with 24 other Flatlanders recieving the test. That would mean we should that 5 out of the 25 of you (because of constraint 1) would actually have the disease.

But, another 5 of your group would not have the Lumps but would anyway recieve a positive test (because of constraint 3). The other 15 people would recieve an accurate negative result.

You are among the 10 people who have a positive result. 5 of the people who tested positive actually have the disease and 5 don’t. So, the probability you tested positive and actually have the diease is 5/10 or 50%.

Conclusion

The conclusion of this puzzle is shocking, the chance you have the disease if you tested positive is no more than toss of a coin! The test, which in hindsight, seemed sound, with a small margin or error, is actually not that accurate! Also, in Flatland(and in our world too), a mere 50% chance of disease is not enough to make a person quaratine themselves.

Another factor would be needed to increase the probability, like people with Lumps have a fever or get a rash. I encourage you to explore these posibilites and find similar intriguing probability puzzles to explore.