Social Science Statistics Blog

Break the problem in two:
a) 100 students take a class, and 50 pass.

Assume for the moment that a student passes or fails the class independent of their peers (which is a reasonable assumption for the initial problem, dealing with the failure rate of vehicles.) Let's assume the standard noninformative prior case, that "half a student" passes and "half a student" fails (the Jeffreys prior) and that students are basically identical. Then the posterior distribution of the probability of passing the test is equivalent to a Beta(50.5,50.5) distribution.

b) Given 50 students passed, on average how many enrolled?
The number of students enrolled in the class for each one who passed is then 1/p - but the mean of 1/p (in this case, 2.02) is necessarily greater than 1/(the mean of p), 2. So the expected class size must be greater under these assumptions. So roughly 101 students enrolled.

The original authors, however, make a profound overestimation of the average of starting students, choosing a "posterior" distribution that yields a class size of 150. To get an expectation this big with this prior information, we would observe a posterior of Beta(2.0,2.0) - or, 1.5 students passing and 1.5 failing! Putting this in perspective, the most likely way I can see this happening is that students pooled their talents and produced 3 distinct final papers: one good, one bad, and one just good enough to get the professor to flip a coin.

It does, however, seem to explain why Harvard classrooms always seem to overflow chaotically at the beginning of each term.

P.S. The original authors call this the "backwards reasoning fallacy", even though Google says the name is better applied to startling schoolchildren deterministically rather than failing them stochastically. Resolving the namespace collision here, does this problem go by another name, or shall we go via Stigler and call it Gelman's paradox?