How Do You Calculate Steady-State Probabilities of a Markov Chain?

The steady-state probability vector is found by solving $T\mathbf{s}=\mathbf{s}$, together with the condition that its probabilities sum to $1$. For a regular Markov chain, repeated multiplication by the transition matrix also converges to this same vector.

The Mechanism

In IB Mathematics: Applications and Interpretation, the usual convention is that $T_{ij}$ represents the probability of moving from state $j$ to state $i$. Therefore, the columns of $T$ sum to $1$, and

$$\mathbf{s}_{n}=T^{n}\mathbf{s}_{0}.$$

At steady state, the probabilities no longer change, so $T\mathbf{s}=\mathbf{s}$.

The extra condition $a+b=1$ is essential because $T\mathbf{s}=\mathbf{s}$ alone produces infinitely many scalar multiples of the eigenvector. Normalizing converts that eigenvector into a probability vector, whose entries are non-negative and total $1$.

Consider

$$T=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix}, \qquad \mathbf{s}=\begin{pmatrix}a\\b\end{pmatrix}.$$

Then

$$\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix} \begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}a\\b\end{pmatrix}.$$

This gives $0.8a+0.3b=a$, together with $a+b=1$. Solving these simultaneously gives

$$a=0.6,\qquad b=0.4.$$

Hence the steady-state vector is

$$\mathbf{s}=\begin{pmatrix}0.6\\0.4\end{pmatrix}.$$

In the long term, $60\%$ of the population is expected to be in the first state and $40\%$ in the second.

On a GDC, either solve the simultaneous equations or calculate a large power such as $T^{50}$ and multiply by $\mathbf{s}_0$. If successive outputs agree to the required precision, the distribution has numerically stabilized. Keep full calculator precision and round only the final probabilities.

Exam Technique

This is HL-only content in AHL 4.19: Transition matrices and Markov chains, examinable in any HL GDC-active paper style. Show the equation or GDC setup before the answer, interpret the probabilities in context, and follow the transition convention stated in the question.

A common misconception is to multiply in the wrong order or transpose the matrix. Under the IB column convention, use $T^{n}\mathbf{s}_{0}$, and check that every column of $T$ sums to $1$.