Question

In a small pond there are five lily pads in a circle labeled 1 through 5. A frog is sitting on pad 1.
When the frog is on pad n, it will jump to right pad with probability 1/2 and to left pad with probability 1/2.
Each jump is independent of the previous jumps.
What is the probability that the frog will return to pad 1 with jumping 7 times?
What is the probability that the frog will return to pad 1 with jumping n times? n -> limit

Answer 1

	1	2	3	4	5
1	0	0.5	0	0	0.5
2	0.5	0	0.5	0	0
3	0	05	0	0.5	0
4	0	0	0.5	0	0.5
5	0.5	0	0	0.5	0

transition matrix be P

the probability that the frog will return to pad 1 with jumping 7 times is element (1,1) in P^7

0.109	0.273	0.172	0.172	0.273
0.273	0.109	0.273	0.172	0.172
0.172	0.273	0.109	0.273	0.172
0.172	0.172	0.273	0.109	0.273
0.273	0.172	0.172	0.273	0.109

Hence, probability that the frog will return to pad 1 with jumping 7 times = 0.109

the probability that the frog will return to pad 1 with jumping n times? n -> limit is when the transition matrix become same

P(n) is

0.200	0.200	0.200	0.200	0.200
0.200	0.200	0.200	0.200	0.200
0.200	0.200	0.200	0.200	0.200
0.200	0.200	0.200	0.200	0.200
0.200	0.200	0.200	0.200	0.200

the probability that the frog will return to pad 1 with jumping n times? n -> limit = 0.2