MIT Mystery Hunt 2020/Food Court

Food Court is a mathematics puzzle from the round of the 2020 MIT Mystery Hunt. It [INSERT BRIEF DESCRIPTION]

Solve Path
The puzzle body consists of twelve probability-based math problems and two directed graphs, along with some flavortext.

Naturally, the first thing to do is to solve all the math problems, obtaining a probability from each.

p1 requires knowledge of Bertrand's ballot theorem, resulting in an answer of (15-4)/(15+4) = 11/19.

p2 can be derived into a geometric series with a = 1/8 and r = 1/8, for p2 = 1/7.

p3 involves taking the complement of an event occuring with probability 6*5*4*3/6^4, for an answer of 1-5/18 = 13/18.

p4 can be visualized along a line segment of length L. p4's event occurs if the cut occurs past (8/9)L or before (1/9)L, for a total probability of 2/9.

p5 is just the Monty Hall problem, with a well-known answer of 2/3.

p6 is a conditional probability problem, with answer (1/10)/(1/10 + 2/5 + 3/5) = 1/11.

p7 requires knowledge of the behavior of an M/M/1 queue. The equation in question results in ρ = 1/2 and π2 = 1/8.

p8 can be simulated as a Markov chain, but the result can also be found here.

p9 is a Nash equilibrium problem.

p10 is an extinction problem, a kind of branching process.

p11 is another conditional probability problem with somewhat different framing.

p12 is a simple example of the inclusion-exclusion principle.

Having obtained all the probabilities, the solver can now turn their attention to the graphs. These can be identified as Markov chains; after plugging in the probabilities, the solver can now solve for the steady state in said Markov chains.

Each of the probabilities in said steady state can be exactly expressed with two decimal digits of precision (in other words, they are integer multiples of 1/100); moreover, they all land between 0.01 and 0.26. Converting to letters yields a cluephrase that yields the answer.