Markov chains and transition matrices (IB Math AI HL)
Updated October 2026
Quick answer: In IB Maths AI HL the transition matrix T holds the probabilities of moving from each state in its columns, so each column adds up to 1. If s₀ is the initial state vector, the state after n steps is Tⁿs₀. The steady state is the vector s with Ts = s whose entries add up to 1: the eigenvector of T for eigenvalue 1.
Weather forecasting, consumer brand-switching, disease spread. HL AI's Markov questions ask for the state after n steps and the steady-state distribution: the eigenvector of the transition matrix for eigenvalue 1.
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Column j of the transition matrix T holds the probabilities of moving from state j to each state, so every column adds up to 1.
The state vector after n steps is sₙ = Tⁿs₀, where s₀ is the initial state. Count the steps carefully: from week 0 to week 3 is T³.
Steady state: solve Ts = s with the entries of s adding up to 1. For a regular chain you can also raise T to a large power on your GDC: every column of Tⁿ gets close to s.
λ = 1 is always an eigenvalue of a transition matrix, and the steady state is its eigenvector, scaled so that its entries add up to 1.
For a regular Markov chain the long-term probabilities do not depend on the initial state.
Worked example
If it is sunny today, the probability that it is sunny tomorrow is 0.8. If it is rainy today, the probability that it is sunny tomorrow is 0.4. Today is sunny. Find the probability that it is sunny in 2 days, and the long-term proportion of sunny days.
With the states in the order (sunny, rainy), the columns are 'from sunny' and 'from rainy': T = [[0.8, 0.4], [0.2, 0.6]], and s₀ = (1, 0).
T²s₀ is the first column of T²: 0.8 × 0.8 + 0.4 × 0.2 = 0.72 sunny and 0.2 × 0.8 + 0.6 × 0.2 = 0.28 rainy.
Steady state (s, r): 0.8s + 0.4r = s gives 0.4r = 0.2s, so s = 2r.
With s + r = 1: s = 2/3 and r = 1/3.
Answer: P(sunny in 2 days) = 0.72. In the long term 2/3 of days are sunny.
5 quick questions on markov chains. No sign-in needed.
A Markov chain has states A and B and transition matrix \(\mathbf{T} = \begin{pmatrix}0.8 & 0.6 \\ 0.2 & 0.4\end{pmatrix}\) (each column is the state it moves FROM). In the long run, what proportion of the time is spent in state A?
A Markov chain has states A and B and transition matrix \(\mathbf{T} = \begin{pmatrix}0.6 & 0.1 \\ 0.4 & 0.9\end{pmatrix}\) (each column is the state it moves FROM). In the long run, what proportion of the time is spent in state A?
A Markov chain has states A and B and transition matrix \(\mathbf{T} = \begin{pmatrix}0.7 & 0.2 \\ 0.3 & 0.8\end{pmatrix}\) (each column is the state it moves FROM). In the long run, what proportion of the time is spent in state A?
A Markov chain has states A and B and transition matrix \(\mathbf{T} = \begin{pmatrix}0.8 & 0.3 \\ 0.2 & 0.7\end{pmatrix}\) (each column is the state it moves FROM). Today \(P(\text{A}) = 0.5\). Find \(P(\text{A})\) in two days' time.
A Markov chain has states A and B and transition matrix \(\mathbf{T} = \begin{pmatrix}0.9 & 0.4 \\ 0.1 & 0.6\end{pmatrix}\) (each column is the state it moves FROM). Today \(P(\text{A}) = 0\). Find \(P(\text{A})\) in two days' time.
Where can I practise markov chains and transition matrices questions?
On the Markov Chains and Transition Matrices questions page: 13 exam-style HL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for markov chains and transition matrices?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working. The Matrices guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.