What does $S_1$ mean in a Markov chain example?

Conditions

• Two-state chain with states A and B.
• Entries are probabilities.
• In the examples shown, the entries sum to 1.

Reasoning, step by step

1. Identify the initial state vector $S_0$.
2. Apply the transition probabilities from the diagram to determine the probabilities of being in each state after one step.
3. Assemble these new probabilities into a column vector, $S_1$.
4. Interpret the top component as the probability of state A and the bottom component as the probability of state B.

Example

The board shows $S_1 = \begin{pmatrix} 0.75 \\ 0.25 \end{pmatrix}$. The speaker explains that 0.75 is the chance of being back in state A after one iteration, and 0.25 is the chance of being in state B.

Common misconceptions

• Thinking $S_1$ represents the state after an arbitrary number of steps.
• Assuming the entries of $S_1$ are the probabilities of transitioning to A or B, rather than the probabilities of being in A or B.

Watch the explanation

Explore next

• Why introduce a matrix instead of just reading the transition diagram?
• How do you compute the next state vector from the current one?