Question

Assume that the following pattern holds true through future generations in a certain country: 60% of the daughters of working women also work and 20% of the daughters of non–working women work. Find the transition matrix of the Markov chain modeling working/non–working states across generations.

Answer 1

Assume two states in our model - working (State 1) and non-working (State 2). So our transition matrix is going to be a 2 by 2 square matrix.

Where represents the probability that State 1 remains State 1 in the next generation, in other words, probability that daughter of a working woman is working. So as given in the question, it is 60%, i.e., a probability of 0.6.

represents the probability that State 1 converts to State 2 in the next generation, in other words, probability that daughter of a working woman is not working. It is nothing but the subtraction of the daughters who are working from the total number of daughters whose mothers work. Hence,

represents the probability that State 2 converts to State 1 in the next generation, in other words, probability that daughter of a non-working woman starts working. So as given in the question, it is 20%, i.e., a probability of 0.2.

represents the probability that State 2 remains State 2 in the next generation, in other words, probability that daughter of a non-working woman is still not working. It is nothing but the subtraction of the daughters who are working from the total number of daughters whose mothers don't work. Hence,

So the transition matrix of the Markov chain modeling the working and non-working states across generations can be obtained by substituting the above calculated values in the matrix.