Question

Assume workers transition through the labor force independently with the
transitions following a homogeneous Markov chain with three states:
• Employed full-time
• Employed part-time
• Unemployed
The transition matrix is:


0.90 0.07 0.03
0.05 0.80 0.15
0.15 0.15 0.70

.
• Worker Y is currently employed full-time
• Worker Z is currently employed part-time
Find the probability that either Y or Z, but not both will be unemployed after two transitions.

Answer 1

Let 0,1,2 repectively denote the transition states Employed full-time, Employed part-time and Umemployed.

Define = state of worker Y at time t and = state of worker Z at time t, t = 0,1,2,..

Note that, are independent. As, we assume workers transition through the labor force independently.

The transition matrix is given by,

. In general, where

Define A as the event that Y will be unemployed(at state 2) after two transitions given he is currently(at time t, say) employed full-time(at state 0) and B as the event that Z will be unemployed(at state 2) after two transitions given he is currently(at time t, say) employed part-time(at state 1), i.e.,

. Note A and B are independent as are independent.

Now,

Similarly,

To find the probability that exactly one of A and B occur, i.e,

[Since, A and B are independent,