

This IB Math AI HL 4.19 – Transition Matrices resource develops students’ understanding of how Markov chains model systems that evolve over time. Students learn to interpret transition matrices, state vectors, and multi-step transitions using powers of the matrix, building a clear link between matrix operations and probabilistic movement between states. The material also develops the concept of steady state (long-term) distributions and how these describe stable long-run behaviour in real contexts.
Structured tasks guide learners through reading transition matrices, modelling two-state systems, calculating future states, and solving steady-state equations exactly. HL extension problems develop higher-level reasoning about regular versus non-regular chains, convergence behaviour, and multi-state models. With applied contexts such as customer switching and subscription tiers, plus a full answer key, this resource supports HL teaching, IA preparation, and deeper understanding of dynamic probabilistic systems in line with IB Mathematics AI expectations.
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