Puzzle 72 | Diseases and Tests

Dinoo is worried that he might have a rare disease. He decides to get himself  tested, and suppose that the testing methods for this disease are correct 99 percent of the time (in other words, if he has the disease, it shows that he does with 99 percent probability, and if he doesn’t have the disease, it shows that he does not with 99 percent probability). Suppose this disease is actually quite rare, occurring randomly in the general population in only one of every 10,000 people.
If his test results come back positive, what are his chances that he actually have the disease?

1. .99
2.  .90,
3. .10
4. .01

Solution: The answer is (d), less than 1 percent chance that he has the disease.

Explanation: 
After discussing the reasons for the surprising probability (below), you should see how changing the parameters affects the outcome. Would the result be so surprising if the disease were more common? How would the probability change if you allow the percentage of false positives and false negatives to be different?

This fact may be deduced using something called Bayes’ theorem, which helps us find the probability of event A given event B, written P(A|B), in terms of the probability of B given A, written P(B|A), and the probabilities of A and B:

                                  P(A|B) = P(A)P(B|A) / P(B)
                                  => P(B) = P(A)P(B|A)/P(A/B)

• In this case, event A is the event he has this disease, and event B is the event that he tests positive.
• Thus P(B|not A) is the probability of a “false positive”: that he tests positive even though he doesn’t have the disease. Here, P(B|A)=.99, P(A)=.0001, and P(B) may be derived by conditioning on whether event A does or does not occur:
  P(B)=P(B|A)P(A)+P(B|not A)P(not A) OR .99*.0001+.01*.9999. Thus the ratio you get from Bayes’ Theorem is less than 1 percent.

The basic reason we get such a surprising result is because the disease is so rare that the number of false positives greatly outnumbers the people who truly have the disease. This can be seen by thinking about what we can expect in 1 million cases. In those million, about 100 will have the disease, and about 99 of those cases will be correctly diagnosed as having it. Otherwise about 999,900 of the million will not have the disease, but of those cases about 9999 of those will be false positives (test results that are positive because of errors). So, if he tests positive, then the likelihood that he actually have the disease is about 99/(99+9999), which gives the same fraction as above, approximately .0098 or less than 1 percent!

Quiz 

This puzzle is contributed by Praveer Satyam.