Given a Markov chain with transition matrix P, we want to find the probability that the grandson of a man from Harvard went to Harvard, where the transition matrix P is:

P = \[\begin{bmatrix} 1 & 0 & 0 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.1 & 0.7 \end{bmatrix}\]

Solution

To calculate the probability that the grandson of a man from Harvard went to Harvard, we square the matrix P. The required probability is the value in the first row and first column of \(P^2\)

Given the transition matrix \(P\) from Example 11.7, to find the probability that a grandson of a Harvard man also went to Harvard, we would:

Here’s the manual calculation:

\[ P^2 = \left[ \begin{array}{ccc} (1 \cdot 1) + (0 \cdot 0.3) + (0 \cdot 0.2) & (1 \cdot 0) + (0 \cdot 0.4) + (0 \cdot 0.1) & (1 \cdot 0) + (0 \cdot 0.3) + (0 \cdot 0.7) \\ (0.3 \cdot 1) + (0.4 \cdot 0.3) + (0.3 \cdot 0.2) & (0.3 \cdot 0) + (0.4 \cdot 0.4) + (0.3 \cdot 0.1) & (0.3 \cdot 0) + (0.4 \cdot 0.3) + (0.3 \cdot 0.7) \\ (0.2 \cdot 1) + (0.1 \cdot 0.3) + (0.7 \cdot 0.2) & (0.2 \cdot 0) + (0.1 \cdot 0.4) + (0.7 \cdot 0.1) & (0.2 \cdot 0) + (0.1 \cdot 0.3) + (0.7 \cdot 0.7) \end{array} \right] \]

Since the first column of the second row and third row in the matrix \(P\) are all zeros, the calculation simplifies significantly:

\[ P^2 = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0.48 & 0.19 & 0.33 \\ 0.37 & 0.11 & 0.52 \end{array} \right] \]

(Where the asterisks “*” represent entries we’re not interested in for this particular calculation.)

Thus, \(P^2\)’s first row and first column entry, which is the probability we’re looking for, is simply 1. This means the probability that the grandson of a man from Harvard went to Harvard is 100%.

The reasoning is because once a son from Harvard is always going to Harvard, this creates a certainty, or a probability of 1, that all subsequent generations will also go to Harvard, given the way the transition matrix has been set up in Example 11.7.

Below is the solution in r:

# Transition matrix P
P <- matrix(c(1, 0, 0,
              0.3, 0.4, 0.3,
              0.2, 0.1, 0.7),
            byrow = TRUE, nrow = 3)

# Squaring the matrix P to find the grandson's probability distribution
P_squared <- P %*% P

# Extracting the probability that the grandson of a man from Harvard also went to Harvard
harvard_to_harvard_probability <- P_squared[1, 1]

# Display the squared matrix and the probability
P_squared
##      [,1] [,2] [,3]
## [1,] 1.00 0.00 0.00
## [2,] 0.48 0.19 0.33
## [3,] 0.37 0.11 0.52
harvard_to_harvard_probability
## [1] 1

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