w = work, .n = not work.Census studies from 1960 reveal that in France 70% of the daughters of working women
also work, and 50% of the daughters of nonworking women work.
Assume that this trend remains unchanged from one generation to the next,
(a) Set up a Markov chain by drawing the transition diagram and the transition matrix.
We have: .
The transition matrix is: .
(b) At that time 40% of French women worked.
Determine the percentage of working women in the next two generations.
We have: . at the first generation.
Second generation: .
. . Working women: 58%
Third generation: .
. . Working women: 61.6%
We want the "steady state" vector.(c) In the long run what percentage of French women will work?
This is a vector such that: .
So we have: .
Then we have: . . These two equatons are always equivalent.
The second equation is always: .
Solve the system: .
Eventually, 62.5% of the women will be working.