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Preamble

For this solution I will employ graphical representation of the problem by using by the plotmat() function from the diagram package to illustrate the Markov chains; also I will solve the matrices in R by employing the efficient operator %^% from the expm package that is used to raise the MChain matrix to the second and third power.

Solution

• We have \(P^1\) (represents the Grandfather’s Son) as follows:

The transition matrix is:

MChain

##           Harvard Yale Dartmouth
## Harvard       1.0  0.0       0.0
## Yale          0.3  0.4       0.3
## Dartmouth     0.2  0.1       0.7

• We have \(P^2\) (represents the Grandfather’s Grand son) as follows:

MChain2 <- MChain %^% 2

##           Harvard Yale Dartmouth
## Harvard      1.00 0.00      0.00
## Yale         0.48 0.19      0.33
## Dartmouth    0.37 0.11      0.52

• We have \(P^3\) (represents the Grandfather’s Great Grandson) as follows:

MChain3 <- MChain %^% 3

##           Harvard  Yale Dartmouth
## Harvard     1.000 0.000     0.000
## Yale        0.603 0.109     0.288
## Dartmouth   0.507 0.096     0.397

Answer:

The probability that the grandson of a man from Harvard went to Harvard is \(P^2\) = 1 or \(100\%\).