Slug on a Triangle

Suppose we have a triangle with vertices A,B,C and a slug that travels around its perimeter. Whenever the slug reaches a vertex, it has a 1/2 chance of moving toward each of the remaining two vertices.

Suppose the slug starts out at vertex A. After the slug has moved 10 triangle side lengths, what is the probability that the slug will find himself back at point A? After moving 20 side lengths, what is the probability that he will be at point C?

after 10 lengths I make it 171/512

after 20 lengths I make it 174763/524288

Originally posted by aging blitzer
after 10 lengths I make it 171/512

after 20 lengths I make it 174763/524288

I agree with your first answer -- well done.

For the second question, I got something very slightly different. I get that your answer would be the probability for the slug to end up at A after 20 moves -- not C. I could have made a mistake, but notice that your second answer is greater than 1/3, which at first glance should be a strike against its being correct.

as you surmised, I didn't read it right.

to be at C after 20
349525/1048576

edit:
either 1 or 2 points will have a greater than 1/3 probability.
after an odd number of moves A < 1/3, B > 1/3, C > 1/3
after an even number of moves A > 1/3, B < 1/3, C < 1/3
so, yes, after 20 the probability of C should be < 1/3

Originally posted by aging blitzer
as you surmised, I didn't read it right.

to be at C after 20
349525/1048576

edit:
either 1 or 2 points will have a greater than 1/3 probability.
after an odd number of moves A < 1/3, B > 1/3, C > 1/3
after an even number of moves A > 1/3, B < 1/3, C < 1/3
so, yes, after 20 the probability of C should be < 1/3

Let An, Bn, and Cn denote the chance of being in A, B and C respectively after n moves.

A0 = 1, B0 = C0 = 0

An = B(n-1)

Bn = 1/2 * (1 - An)

Cn = Bn

I currently have no program to check the answers given 🙂

Originally posted by aging blitzer
as you surmised, I didn't read it right.

to be at C after 20
349525/1048576

edit:
either 1 or 2 points will have a greater than 1/3 probability.
after an odd number of moves A < 1/3, B > 1/3, C > 1/3
after an even number of moves A > 1/3, B < 1/3, C < 1/3
so, yes, after 20 the probability of C should be < 1/3

Well done.