The student has demonstrated a consistent and correct use of mathematical notation, symbols, and terminology throughout the work. Key terms and variables were appropriately defined and explained when first introduced, and the main concepts were effectively described in the introduction, particularly the Markov chains. Multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs, and models, were present and used appropriately. However, the student should have provided more detailed explanations for the mathematical calculations and data presentation, particularly regarding the matrices. Additionally, when rounding values, the student should have stated the degree of accuracy. Overall, the student effectively used deductive methods and logical proofs where appropriate and appropriately labeled graphs, tables, and diagrams.