Research task (STEM):
Write an introduction on Markov chains in the field of statistics/combinatorics and the reason behind their genesis. Then, write a literature review on Markov chains in contemporary research. Include one section devoted to their application and intuition in solving problems like "what is the expected value of flips to get N heads in a row". 

Evaluation rubric:
criterion: The response includes sections for introduction of Markov chain, applications of Markov Chain, and intuition behind N heads problem.
weight: 4
axis: Communication Quality

criterion: The response provides an accurate definition of Markov chain which is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the previous states.
weight: 5
axis: Explicit Criteria

criterion: The introduction mentions that Andrey Markov invented Markov chains in the early 1900s.
weight: 5
axis: Explicit Criteria

criterion: The introduction references Markov's 1913 work on analysis of the sequence of vowels and consonants in Alexander Pushkin's novel, Eugene Onigin.
weight: 4
axis: References & Citation Quality

criterion: The response discusses at least three distinct modern research areas (e.g. Markov Chain Monte Carlo, Markov Decision Processes/Reinforcement Learning, Mixing Times/Random Walks on Graphs, Queuing Theory, Computational Biology, Finance, Statistical Physics).
weight: 4
axis: Explicit Criteria

criterion: All citations are in an official format (e.g. Modern Language Association, American Psychological Association, Chicago, IEEE).
weight: 1
axis: References & Citation Quality

criterion: The response includes an intuitive explanation of how to solve the coin-flip problem (e.g. description of states representing the current run length of consecutive heads (0 through N), a tail resets the run length to 0, example scenario where N is a certain number like 3).
weight: 4
axis: Explicit Criteria

criterion: The response states that for a fair coin, the expected number of flips needed to get N consecutive heads equals 2^{N+1} − 2 or an algebraically identical formula.
weight: 5
axis: Explicit Criteria

criterion: The response includes out of scope contents, such as  moderation policies, ethics statements, ethics in gambling, and other unrelated probability theorems.
weight: -4
axis: Synthesis of Information

criterion: The response discusses the reason behind the genesis of Markov chain which was either the need to model dependent events or extend law of large numbers to correlated trials.
weight: 4
axis: Explicit Criteria

criterion: The response includes tangential topics unrelated to Markov chains (e.g. differential equations, unrelated probability models, graph theory, number theory).
weight: -2
axis: Synthesis of Information

criterion: All mathematical statements and calculations are correct and aligns with basic probability facts.
weight: 3
axis: Implicit Criteria

criterion: The response discusses how Markov chains are used with linear algebra (e.g. transition matrix, eigenvalues, state probability vectors, time evolution).
weight: 3
axis: Implicit Criteria

criterion: The response discusses key properties related to Markov chains (e.g. memoryless, irreducibility, aperiodicity, ergodicity).
weight: 2
axis: Synthesis of Information

criterion: The response contains at least one problem similar to the coin-flip problem specified in the prompt.
weight: 2
axis: Implicit Criteria

criterion: The response includes mathematical foundations of the Markov chain (e.g. state space, transition matrix, transition probabilities, initial distribution, stationary distribution).
weight: 4
axis: Explicit Criteria

criterion: All mathematical equations are in LaTex format.
weight: 2
axis: Communication Quality

criterion: The response discusses at least one non-scientific domain where Markov chain is used such as philosophical dispute over dependence and free will.
weight: 1
axis: Implicit Criteria

criterion: The response discusses key variations of Markov chains (e.g. finite vs infinite state space, time-homogeneous, high-order chains, discrete-time vs continuous-time).
weight: 2
axis: Synthesis of Information

criterion: The response explains how first-step analysis is used to solve N heads in a row problem.
weight: 2
axis: Implicit Criteria