Chapter 9.2, Problem 68E

Politics: filibuster. The Senate is in the middle of a floor debate, and a filibuster is threatened. Senator Hanks, who is still vacillating, has a probability of $.1$ of changing his mind during the next $5$ minutes. If this pattern continues for each $5$ minutes that the debate continues and if a $24$ -hour filibuster takes place before a vote is taken, what is the probability that Senator Hanks will cast a yes vote? A no vote?

(A) Complete the following transition matrix:

$\begin{array}{l}\text{ Next}\text{5 minutes}\\ \begin{array}{cc}Yes& No\end{array}\\ \begin{array}{c}\text{current}\\ \text{5 minutes}\end{array}\begin{array}{c}Yes\\ No\end{array}\left[\begin{array}{cc}.9& .1\\ & \end{array}\right]\end{array}$

(B) Find the stationary matrix and answer the two questions.

(C) What is the stationary matrix if the probability of Senator Hanks changing his mind ( $.1$ ) is replaced with an arbitrary probability $p$ ?