Hint to Riddle #66: Disease Testing Accuracy.

66. A disease called The Phage spreading throughout a kingdom of 1,000,000 people. Currently one in every 500 people have the disease.
The King's scientists develop a test with accuracy as follows: It will fail by false negative at a 1% rate. It will fail by false positive at a 2% rate. The King, planning to test everyone in the kingdom, is pleased. He thinks if he was prepared to secretly murder the 2,000 infected people a 2% rise wont make any difference. 'Hold on' says the chief nerd 'you'll actually have to murder X people and only one in Y of them will have The Phage.'

What are X & Y?
Real life problem this the understanding of which can prevent serious problems. The answer can be quite surprising. I'd suggest you just run the numbers and have confidence. Also read the paragraph below to make sure you are clear on the nomenclature.

False Negative & False Positive

This terminology is used in all sorts of things that give only two possible outcomes, spam filters, virus tests (computer,) and what we are doing here.

False Positive simply is when someone is given a positive result that shouldn't i.e. the person does not have the virus
False Negative conversely is when someone is given a negative result when they shouldn't i.e. the person has the disease.

Thus, perhaps not completely obviously, True Positive rate is 1 - (False Negative rate) and True Negative rate is 1 - (False Positive rate)... Consider if you are testing 100 people that have the disease there can be no false positives but you will get some false negatives. In our case the false negative rate is 1% or 0.01, so when testing 100 infected people we would get results indicating 1 negative hence 99 positive. So the True Positive rate is 99%. Another way of phrasing the question would be that the test is 99% accurate for positive results and 98% accurate for negative results.

People often concern themselves with false negatives, for example how much spam gets through your spam filter. But really the damage is done by the false positive. Real mail being blocked.

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