I had had an analysis exam the previous day, that I'd done very poorly on, and as I sat to take the probability exam I was exhausted and really couldn't think straight. I hadn't prepared well for it at all, and couldn't do basic things like subtract correctly (even using a calculator) or perform simple algebra tasks, much less think creatively about problems.
I felt reasonably well-prepared for this exam - the last few weeks, the material has seemed to come together for me much more than it did in the middle of the course, and I had good formula sheets written up - but I also worked hard to be rested, correctly fed, hydrated, etc., for the exam. I knew that I would need (because both probability and tests in general demand it) mental flexibility in order to be able to answer the questions.
And I did it. I completely killed the exam - I should have a perfect score, or at least within epsilon of a perfect score. (Really I could have as low as a 95% - who knows what weird errors I could have made - but I definitely got the questions basically correct.) And I didn't just kill the exam by being prepared; I killed it by being smart (relative to my baseline) and mentally flexible.
One question asked us something about three independent random variables, each uniformly distributed on the interval [0,1]. (For the "probabilists" out there: we had to determine the CDF and expected value of the minimum term. Pretty easy stuff.)
The next question asked us to consider the same three variables, and then had some questions that only involved two of the variables. I had a few moments of confusion (of the type that totally derailed me on the secon exam) before realizing that the irrelevance of the third variable meant I could draw the standard [0,1]x[0,1] box and fill in the areas I was being asked about and do the computations using areas (e.g., "What is the probability that Y1 > Y2 given that Y1 > Y2^2" - which is the just the ratio of two areas, given that the distribution is uniform).
The exam wasn't hard either, I should admit, but it did take me just about the entire time.
There were five problems (some with multiple parts) worth 15-20 points each, and then the sixth problem was worth 6 points. He's urged us to consider these last problems as pretty much optional, even though they are part of the full score. If you make sure you can do the basic problems, then it's OK if you can't do the fancy problem. But this fancy problem turned out to be exactly the type of fancy problem that I am good at. It went something like this:
An urn contains 6 red balls and 14 blue balls. Two balls (selected at random) are removed and discarded without noting their colors, and then another ball is drawn. Given that this last ball drawn is red, what is the probability that both of the two discarded balls were blue?This is exactly the kind of thing where if you just draw a little probability tree it is pretty obvious how to compute it. (I shouldn't say it's "obvious." Many things in probability should be obvious but take me a long time to figure out or I can't figure them out at all. But this type of problem is intuitively easy for me, for whatever reason.)
I have my analysis exam tomorrow, which is triple express doom, and then the (low-stakes) logic exam on Wednesday. After that, I have to write a [maximum] 3-page teaching philosophy and fill out a short survey for one of my courses, and then I'm done.