Tomorrow is the annual meetup for the Northwest Region Marble Lovers Association. The days festivities will include various marble games, admiring collections of marbles, a guest speaker who will talk about the virtues of glass spheroids, and running an impressive Rube Goldberg machine that you've spent the past few months setting up.
The most important part of the meeting, though, is the cake. The cake (chocolate-marble, of course) has a single marble dropped in the batter before baking. Out of the 12 attendees, whoever gets the piece with the marble in it will host the next years meeting!
Normally, the host would mark part of the cake pan so that they could know where the marble is, but you're a bit of a klutz, and you dropped the marble into the batter without seeing where it went. The marble has a diameter of half an inch, and the cake pan is round, with a radius of 6 inches.
You realize that, since you don't know where the marble is, there's a chance that you'll run into it while trying to cut the cake. You don't want to remove the batter and start over - that would mess up the marbling of the cake, and be a waste of perfectly good ingredients! You decide to make a second cake, though it may be a bit of a rush to finish it on time.
The only other pan you have left is a rectangular 9 inch by 13 inch cake pan. Fortunately, that pan is roughly the same area, so the original recipe still works. You make up a second set of batter, add it to the pan, and are about to place the marble when -oops! - you drop this marble too! The event is getting close, so you decide to bake both cakes and hope for the best.
Now you have two delicious cakes, but don't know where the marble is in either of them. You want to minimize the probability of slicing into the marble while serving the rest of the guests. The only requirements are that all cuts are perfectly straight, and that in the end there are 12 equal area pieces, though the pieces do not have to be congruent. You can assume that the knife is perfectly sharp and that cuts have no width. You may trim parts of the cake off, but trimmed pieces can't be large enough to potentially contain the marble. You must take the cuts required to trim the cake into account in making your probability calculations.
What is the optional strategy for cutting each of the two cakes? Which cake offers the best overall chances of avoiding the marble?
I myself do not know what the optimal strategy is, or really how to prove it, which I should have stated when I first posted the question. Because of this, to keep the question from staying open indefinitely, the answer with the best chances of avoiding the marble was accepted on October 27th, 2017
The best strategy found was
Cutting the rectangular cake into a 2x6 grid of rectangular pieces, with only a 22.35% chance of hitting the marble.
This answer was provided by both Penguino and Nopalaa. However, Penguino found a better way to cut the circular cake, so overall provided the best chances of avoiding humiliation. Both showed good attention to detail in how they answered the questions and showed how they arrived at their probabilities.