Exercise 11.1 #4
For Example 11.6, find the probability that the grandson of a man from Harvard went to Harvard.
Given the following Markov chain from the example:
\(P=\begin{matrix} \begin{matrix} \\ H \end{matrix} \\ Y \\ D \end{matrix}\begin{matrix} H & Y & D \\ \begin{matrix} .8 \\ .3 \\ .2 \end{matrix} & \begin{matrix} .2 \\ .4 \\ .1 \end{matrix} & \begin{matrix} 0 \\ .3 \\ .7 \end{matrix} \end{matrix}\)
Since we are looking for the probability of the grandson, who is the son of the son, then weโre looking for \({ p }_{ HH }^{ (2) }\).
This will be equal to
\(={ p }_{ HH }\times { p }_{ HH }+{ p }_{ HY }\times { p }_{ YH }+{ p }_{ HD }\times { p }_{ DH }=(.8)(.8)+(.2)(.3)+(0)(.2)=.7\)
(.8*.8)+(.2*.3)+(0*.2)## [1] 0.7