The nation of Griddonesia consists of eightyone equallyspace

The nation of Griddonesia consists of eighty-one equally-spaced islands depicted by the intersections of the lines in the grid below. The lines themselves represent horizontal and vertical bridges exactly one-mile long that connect the islands. A Griddonesian environmental engineer named Hilo has designed a small flying robot for the continuous monitoring of air-pollution levels on the islands. The Robird® is programmed as follows. After taking a pollution reading on an island, it is equally likely to fly to an other island for the next reading. The process then repeats automatically. If the Robird® starts on the NE island, what is the probability pn that after exactly n flights, it is on the SW island? Formulate and solve an appropriate difference equation.

Solution

let pn denote the probability that the robird returns to the SW island after n flights,
where p0 = 1
Using the the law of total probability :
pn+1 = P(on SW after n + 1 flights) = P(on SW after n + 1 | on SW after n)P(on SW after n) +
                                          P(on SW after n + 1 | not on SW after n)P(not on SW after n )

=> P(n+1) = 0 + (1/80)(1 - pn), n = 0, 1, 2, … .

P(n+1) = (1/80)*(1-Pn) , n= 0,1,2,.....

when n = n-1

=> P(n-1+1) = (1/80)*(1-P(n-1))
P_n = (1/80)*(1 - P_(n-1)))

P₀ = 1