Question:
Consider a Markov Chain with state space \(S = \{0, 1\}\) and the transition probability matrix \(P = \begin{pmatrix} 1 & 0 \\ 0.5 & 0.5 \end{pmatrix}\). What is the value of \(P^{(2)}_{10}\)?
Consider a Markov Chain with state space \(S = \{0, 1\}\) and the transition probability matrix \(P = \begin{pmatrix} 1 & 0 \\ 0.5 & 0.5 \end{pmatrix}\). What is the value of \(P^{(2)}_{10}\)?
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Use the Chapman-Kolmogorov relation \(P^{(2)}_{10}=\sum_k P_{1k}P_{k0}\), or simply square the given 2 by 2 matrix.
Updated On: Jul 4, 2026
- 1
- 0.5
- 0.75
- 0
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The Correct Option is C
Solution and Explanation
Step 1: The two step transition probability \(P^{(2)}_{10}\) is the \((1,0)\) entry of \(P^2\), given by the Chapman-Kolmogorov equation \(P^{(2)}_{10} = \sum_{k \in S} P_{1k}P_{k0}\).
Step 2: Here \(S=\{0,1\}\), so \(P^{(2)}_{10} = P_{10}P_{00} + P_{11}P_{10}\). From the given matrix, \(P_{00}=1,\; P_{01}=0,\; P_{10}=0.5,\; P_{11}=0.5\).
Step 3: Substitute the values: \(P^{(2)}_{10} = (0.5)(1) + (0.5)(0.5) = 0.5 + 0.25 = 0.75\).
Answer: option (C), 0.75.
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