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. :)
Answer
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.
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