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Linear AlgebraEasyMarkov Chains & Long-Run Behavior

Markov Chains & Long-Run Behavior · Easy problem

A two-state chain has transition matrix M=(0.60.10.40.9)M = \begin{pmatrix} 0.6 & 0.1 \\ 0.4 & 0.9 \end{pmatrix}, where MijM_{ij} is the probability of moving to state ii from state jj. The chain starts in state 1 with probability pp and in state 2 with probability 1−p1 - p. After one step the probability of being in state 1 is 0.30.3. What is pp? Express your answer as a decimal.