Question: Harry and the comely young lady. Harry, the famous restaurateur has his eye on a comely young lady, whom he originally spotted at a high cholesterol cooking course. However, during the course Harry never had the courage to introduce himself as a famous restaurateur. The young lady eats lunch regularly on Optima Street where Harry’s restaurant is located and Harry, quite smitten with her, observes that she visits the 4 restaurants on the street labeled 0,1,2, and 3, according to the Markov transition matrix

\[P=\left[ \begin{matrix} 0.1 & 0.2 & 0.7 & 0 \\ 0.2 & 0.3 & 0.3 & 0.2 \\ 0.3 & 0.1 & 0.1 & 0.5 \\ 0.4 & 0.3 & 0.2 & 0.1 \\ \end{matrix} \right]\]

Harry’s restaurant is labeled 0 and the hated sprouts bar across the street is labeled 3. Assume initially she is equally likely to visit any of the 4 restaurants.

(a) What is the probability she visits Harry’s before the hated sprouts bar?

(b) What is the expected waiting time until she visits Harry’s for the first time?

(c) What is the expected waiting time between two visits to Harry’s?