Assume that a man’s profession can be classified as professional, skilled laborer, or unskilled laborer. Assume that, of the sons of professional men, 80% are professional, 10% are skilled laborers, and 10% are unskilled laborers. In the case of sons of skilled laborers, 60% are skilled laborers, 20% are professional, and 20% are unskilled. Finally, in the case of unskilled laborers, 50% of the sons are unskilled laborers, and 25% each are in the other two categories.

Assume that every man has at least one son, and form a Markov chain by following the profession of a randomly chosen son of a given family through several generations. Set up the matrix of transition probabilities. Find the probability that a randomly chosen grandson of an unskilled laborer is a professional man.

p <- matrix(c(0.8, 0.1, 0.1,
              0.2, 0.6, 0.2,
              0.25, 0.25, 0.5), nrow = 3, byrow = TRUE)
rownames(p) <- c("Professional", "Skilled", "Unskilled")
colnames(p) <- c("Professional", "Skilled", "Unskilled")
p
##              Professional Skilled Unskilled
## Professional         0.80    0.10       0.1
## Skilled              0.20    0.60       0.2
## Unskilled            0.25    0.25       0.5
p.squared <- p %*% p

p.squared
##              Professional Skilled Unskilled
## Professional        0.685   0.165     0.150
## Skilled             0.330   0.430     0.240
## Unskilled           0.375   0.300     0.325

Given that the matrix is applicable only for children, we need to calculated \(P^2\) for a grandchild. So the probability that a randomly chosen grandson of an unskilled laborer will turn out to be a professional man is only 0.375.