# The Ravages of Purple Spots Disease

While browsing the Internet for medical lore, you happen on a medical encyclopedia article about purple spots disease.

Instances of purple spots disease have been documented in 150 countries. The disease affects one in every 100,000 individuals and there is no known cure. Individuals who have the disease remain completely asymptomatic until one day they very suddenly break out in purple spots that never go away.

Concerned that you may have purple spots disease and not know it, you scour the Internet and locate a reputable medical testing company that provides a screening test for the disease. The test is state of the art. It guarantees 99.5% accuracy, meaning that whether you have the disease or not, the result returned by the test will be correct 99.5% of the time. The test costs $300.00 USD to administer. Figuring that your peace of mind is worth$300.00, you undergo the test for the disease.

Three weeks later, the results come back. The test is positive! You're advised to come in for a retest to be absolutely sure.

Based on the information above, should you be deeply worried? Why or why not?

Puzzlers are politely encouraged to place answers in spoiler blocks to avoid inadvertently spoiling the fun for other readers. :)

No you shouldn't be too worried. The probability you have the disease is 0.199%, or in other words about a 1/500 chance.

First, let us assume there are 10,000,000 people in the society (the number you assume here is irrelevant, and you could even just let x be the number if you were mathematiaclly inclined). Now given that 1/100,000 people have the disease, therefore, 100 people will have it in this society. Using this information a 2 by 2 table can be constructed as seen below:

             Has Disease        No Disease         Total
+ve test        99.5              49999.5          50099
-ve test         0.5            9949900.5        9949901

Total            100              9999900       10000000


From the above table, given a +ve test result, the probability of having the disease can be evaluated by computing by 99.5/50099 = 1.99 x 10^-3, or about a 1/500 chance.

YES! Fear! Panic! Runabout! Diseases are scary! They said so on TV!

Credit where credit's due, Mew beat me to a mathematically correct answer

The full answer is that it depends on the economic cost of being a purple spots disease carrier without knowing it. If you can be held financially responsible for anyone you transmit it to, and it does \$100,000 in damages to anyone whom it infects, it would be fiscally wise to either become a hermit until the purple spots break out, or get retested because the legal process has a 1/500 chance of being REALLY expensive (stealing the 1/500 number from Mew)