Random Pub Crawl: Your town has 5 pubs close to your home, say A, B, C, D, E. 1/4 1/2 B Home 1/2 1/2 1/4 1/4 1/4) 1/2 1/4 1/4 1/3 1/4 1/3 1/2 D E 1/3 1/2 (a) Write a program to simulate the random pub crawl. (From the "home" state allow probability 1/2 of going back to state B). (b) Starting from the Home state, run your program 1000 times, each time simulating a Markov chain of length 100

. Each simulation should output a random sequence of values (81, 82, 83 ..., $100 ) where each s, is a pub and with that sequence, compute the num- ber of times each state is visited. Keep only these statistics. After the 1000 repetitions compute the average number of visits to each state. 3 (c) How do the statistics change if you started at state D? Repeat the simulation staterting at state D. (d) Does the distribution over states visited at step t converge for large t? (e) Approximately how long does it take for the chain to "forget""> whether it started at Home or at D?