Question

(5 A flea moves around the vertices of a triangle in the following manner: Whenever it is at vertex i it moves to its clockwise neighbour vertex with probability p; and to the counterclockwise neighbour with probability qi =1- Pi,i = 1,2,3. (a) Find the proportion of time that the flea is at each of the vertices. (b) How often does the flea make a counterclockwise move which is then followed by five consecutive clockwise moves?

Answer 1

ANSWER Given that dock wise neighbowl vertex with probability B and to the counter clock wise neighbours with probability. q = 1-PI where i = 1,2,3 a) The probability of time that the fiea is at each of the veutices the process is state o. If the fly is at vedtan I State 1. If the fly is at vedtax 2 state 2. If the fly is at vertan 3. The Erars ition probability makrise is given by Palo 11-B © -P , 11-19 0 P. | Pz iPg By using the above toansition probability mautorise limiting probabili tgs are

To = C1-B2) 77, +2,77, T, = P, Hot C1-P₂) T₂ T = (1-P).TTO +2.77 we know that To ti +Tl₂ = 1 solve the above theore equation ur get the vakre of To II, of Tz respectively b) As This is markon - chain it is nemodry less and it does not depend where it was in previous 5 consecutive times Thus for the case, it is same as the probability the two probability that the sicamuts the counter clock wise neighbouor at any given time, which is Pi (1) * CI-PD + Pi (2) * C1-2) +P/13) *(1-P)