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 percent are professional, 10 percent are skilled laborers, and 10 percent are unskilled laborers. In the case of sons of skilled laborers, 60 percent are skilled laborers, 20 percent are professional, and 20 percent are unskilled. Finally, in the case of unskilled laborers, 50 percent of the sons are unskilled laborers, and 25 percent 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.

Let the state \(professional = P\), \(skilled\;laborer=S\) and \(unskilled\;laborer=U\). Given the provided probabilities, the matrix of transition probabilities for one step is as follows:

\[ P= \begin{equation*} \begin{array}{c|ccc} & \text{P} & \text{S} & \text{U} \\ \hline \text{P} & 0.8 & 0.1 & 0.1 \\ \text{S} & 0.2 & 0.6 & 0.2 \\ \text{U} & 0.25 & 0.25 & 0.5 \\ \end{array} \end{equation*} \] Since these probabilities represent the odds of a worker’s son being a certain type of worker (which we can refer to as \(P_{son}\)), \(P_{grandson}\) is given by \(P_{son}\) * \(P_{son}\).

P_son <- matrix(c(0.8, 0.1, 0.1,
                0.2, 0.6, 0.2,
                0.25, 0.25, 0.5), 
               nrow = 3, byrow = TRUE)

P_grandson <- P_son %*% P_son
P_grandson
##       [,1]  [,2]  [,3]
## [1,] 0.685 0.165 0.150
## [2,] 0.330 0.430 0.240
## [3,] 0.375 0.300 0.325

Given these derived probabilities, the probability of an unskilled laborer having a professional grandson is around \(37.5\%\).

Plotting out each path directly confirms this result:

#paths from U
U_to_P <- 0.25
U_to_S <- 0.25
U_to_U <- 0.5

#paths to P
P_to_P <- 0.8
S_to_P <- 0.2
#U_to_P is stored above

U_to_Anything_to_P <- U_to_P*P_to_P + U_to_S*S_to_P + U_to_U*U_to_P

U_to_Anything_to_P
## [1] 0.375