In Episode 207, "Convergence," an old college rival comes to CalSci (Marshall Penfield, played by Colin Hanks,) to present new research on a gap he has found in Charlie's proof. Although Charlie does find a way to resolve the error term, it isn't clear to many people watching still what type of math Charlie really does do! What he studies falls into the area of probability and statistics involving what are called "random matrices." While the first activity has no prerequisites, the latter activities all assume some basic knowledge of linear algebra, matrices, and determinants.
The teacher poses to his students the Monty Hall problem, which investigates the probability of choosing a correct answer given three options and the subsequent probability of changing the original choice. The teacher explains why his students' instincts are not correct.
Charlie explains how he narrows down a suspect pool using reverse decision theory. He compares decision theory to a hunter. You can use decision theory to see what animal a hunter is most likely to kill. You can also use the reverse and look at a kill and determine which hunter was most likely to have hunted that animal.
In this clip, the main mathematical concepts derive from a field called game theory. The goal of game theory is to take some social situation with some notion of winning and come up with a formal, mathematical way of discussing it.