If we have two professors teaching the same course course, and last semester of professor A's students, 63 failed and 2037 succeeded while of professor B's students, 16 failed and 784 succeeded, then the success rate of professor A is 97% and the rate of professor B is 98%. So clearly, if we were would be doing that course the coming semester our professor of choice would be B.
Now we refine our statistics on last semester's results and differentiate on the students that prepared for the exam and those that didn't. A's prepared students failed and succeeded 6 and 594 times respectively, yielding a 99% rate. For B, this is 8 and 592, yielding 98,7...%. Regarding the unprepared students, professor A has 57 failed and 1443 succeeded, giving us 96,2%, and for B this is 8 and 192 resulting in 96%.
With this new information, regardless of whether we're going to prepare for the exam, the best option has become professor A! Intuitively I can't grasp how this is possible: we're still talking about the same student population here. Is there any explanation that makes this result seem more natural, intuitively?
