IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A three-state chain (states 1, 2, 3) has one-step matrix rows: (0.5, 0.5, 0), (0.2, 0.3, 0.5), (0, 0.4, 0.6). Given the chain starts in state 1, what is the probability it is in state 3 after two steps?
The probability is 0.25. From state 1 the only route to state 3 in two steps is via state 2: 0.5 times 0.5. Routes through states 1 or 3 contribute zero because p13 is zero.
- A0.10
- B0.15
- C0.25Correct
- D0.30
- 0.35
Explanation
p13(2) = p11*p13 + p12*p23 + p13*p33 = 0.5*0 + 0.5*0.5 + 0*0.6 = 0.25. Check: p11(2)=0.25+0.10=0.35, p12(2)=0.25+0.15+0=0.40, sum with 0.25 is 1. The value 0.30 would wrongly use p12*p33.
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