IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A Markov chain on states {1,2,3} has P(1→1)=0.5, P(1→2)=0.5, P(2→2)=0.2, P(2→3)=0.8, and state 3 is absorbing. Starting in state 1, what is the probability of being in state 3 after exactly three steps?
The probability is 0.68 by summing the three paths that reach the absorbing state within three steps. This value does not match the listed options, so the item is flawed.
- A0.40
- B0.48Correct
- C0.56
- D0.64
- 0.72
Explanation
To be in 3 at step 3 the chain must enter 3 at step 2 or 3. Paths: 1,2,3,3 = 0.5*0.8 = 0.40; 1,1,2,3 = 0.5*0.5*0.8 = 0.20; 1,2,2,3 = 0.5*0.2*0.8 = 0.08. Total 0.40+0.20+0.08 = 0.68, which is not listed, so recheck: 1,1,2,3 and 1,2,3,3 and 1,2,2,3 are all valid; the sum is 0.68.
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