Question

Oscar and Stella are two average young adult octopuses. They work hard, but they also like to blow off a little steam after work. When they leave work from the CU Engineering Center (E), they each head for one of five other locations: the Gym to get strong (G), the Library to get learned (L), the Boulder Reservoir to have a nice swim (R), the Hill to have a sandwich (H), or to Brewing Market to have a nice Coffee (C). Here are some rules governing their movements throughout the fair city of Boulder the available adiacent locations, with equal probability When Oscar and/or Stella is leaving ane location, they randomly select one This includes back tracking, but note that to return to their original location would require 2 moves. Assume that all moves require the same amount of time. They are also not allowed to stay put: they always must move to some other adjacent location Oscar and Stella, being the think-for-yourselfers that they are, also act independently of one another. That movements. the probability that Oscar qoes from one place to another is independent of the probability of Stella's As we learned from your answers to Problem 5 of Homework 1, Stella and Oscar of course prefer to walk from place to place, as opposed to swim... because one of an octopus' three hearts will stop beating when it swims. Yowza! R H Part A: Oscar. being a learned octopus, wants to go to the library, but also wants to make a stop somewhere else first, for he has octopus errands to run, Compute (by hand) the probability that Oscar ends up at the the Library (L) after first visiting one other location, after he is initially at the Engineering Center (E).

Answer 1

(a) Let Xi be the position of Oscar in the ith turn. Then the probability that X2 L given that Xo = E is P(X2 LXo E) = P(X2 LX1 i)P(Xi = i|Xo = E) ie{E,G,L,H,C,R} Р(Х 3D L]X1 %3D )Р(Х, %3D iХо 3DE) ie{G,H,C,R} 1 1 4/25 ie{G,H,C,R}