IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A Markov chain on states {1, 2, 3} has transition matrix rows: state 1: (0.5, 0.5, 0); state 2: (0.2, 0.3, 0.5); state 3: (0, 0, 1). Starting in state 1, what is the probability that the chain is in state 3 after two steps?
The two-step probability is 0.25. The only route from state 1 to state 3 in two steps goes via state 2, with probability 0.5 for 1 to 2 and 0.5 for 2 to 3, giving 0.25.
- A0.25Correct
- B0.10
- C0.50
- D0.15
- 0.35
Explanation
P^2(1,3) = P(1,1)P(1,3) + P(1,2)P(2,3) + P(1,3)P(3,3) = 0.5x0 + 0.5x0.5 + 0x1 = 0.25. The value 0.50 results from using only the one-step probability of reaching state 2 and ignoring the 0.5 factor for moving on to state 3.
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