The sickness puzzle

A nice puzzle... Read the code below...

#!/usr/bin/env python import random # A sickness with no symptoms has appeared in the general population: # Chances of getting it, are one in ten thousand. # # Whoever has it, dies within a year. # # Naturally, people want to know if they are sick or not - to make sure # their affairs are in order, fix their wills and testaments... say their # farewells to their loved ones, and come to terms with their fate. # # Scientists have devised a test, which can show whether you have the # sickness or not, with an accuracy of 99%. # # You just took the test - and you are told that it is positive. # # Should you start taking care of your post-mortem affairs? # Or is it too soon? # # Don't read the rest, before you think about this. # Be honest - no cheating! # # # # Well, let's assume that we have one million test subjects. # One in ten thousand of them will be sick, which means that... # # 1.000.000 x 1 / 10.000 = 100 sick people # # ...100 of them will be sick - and 999.900 will be healthy. # Now we perform the test on all one million of them: remember, # the test is 99% accurate, so... # # 1.000.000 people # / \ # / \ # / \ # 100 sick 999.900 healthy # /\ / \ # / \ / \ # 99 pos 1 neg 9.999 pos 989.901 neg # # ...out of the 100 sick, it will give a positive result to 99, # and miss only one, who will get a negative. Out of the 999.900 # healthy ones, 1% (that is, 9.999) will get a false positive, # and the rest (999.900 - 9.999 = 989.901) will get a correct, # negative result. # # So, grouping all who got positive results, you have: # # - 99 that are REALLY sick # - and 9.999 that are healthy # # So, given the fact that the result is positive, your chance # of ACTUALLY being sick, is: # # 99 # P(I am sick if my test is positive) = --------- # 99 + 9999 # # ...which amounts to a little less than 1%. # In other words, keep partying! # # # The above is a simple example of the Bayes theorem, whose # formal application is the following: # ( the symbol '|' means 'given' ) # # P(Sick)*P(TestPositive|Sick) # P(Sick|TestPositive) = ------------------------------ # P(TestPositive) # # P(Sick)*P(TestPositive|Sick) # = ----------------------------------------------------------------- # P(Sick)*P(TestPositive|Sick) + P(Healthy)*P(TestPositive|Healthy) # # (1/10000)*0.99 # = ------------------------------------- = 0.0098039215 # (1/10000)*0.99 + (9999/10000)*0.01 # # What follows below, is an experimental verification, # via a simple Python reproduction of the experiment. # First, a helper function: # When passed a number between 0 and 100, this function # returns True with exactly the requested chance # e.g. BoolProb(40) would return True with a 40% chance # def BoolProb(x): return random.random() < x/100. # And now, the complete experiment: # The experiment's sample size SampleSize = 10000000 # How many positive results we got PositiveTests = 0 # How many of the positive results were actually sick PositiveTestAndSick = 0 for i in xrange(0, SampleSize): sick = BoolProb(0.01) # 1 in 10.000 => 0.01 if sick: # If you are sick, you get a 99% chance of a positive testResultPositive = BoolProb(99) else: # If you are healthy, you get a 1% chance of a positive testResultPositive = BoolProb(1) # Count the results if testResultPositive: PositiveTests += 1 if sick: PositiveTestAndSick += 1 print "Chance of being sick if test is positive:", \ float(PositiveTestAndSick)/PositiveTests

...that is, less than one per cent chance.bash$python ./Am.I.sick.py Chance of being sick if test is positive: 0.00955057847804

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