A random walk on the 2-dimensional integer lattice begins at the origin. At each step, the walker moves one unit either left, right, or up, each with probability 1/3. (No downward steps ever.)

A walk is a success if it reaches the point (1, 1).

What is the probability of success?

The 9 combinations that exist for two movements:

R,R
R,L
R,U *
L,R
L,L
L,U
U,R *
U,L
U,U

Walker would only be able to make it to (1,1) with only 2 of these combinations.

So if he were only allowed to make two movements, his chances of arriving at (1,1) would be 2/9.

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A random walk on the 2-dimensional integer lattice begins at the origin. At each step, the walker moves one unit either left, right, or up, each with probability 1/3. (No downward steps ever.) A walk is a success if it reaches the point (1, 1). What is the probability of success?

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A random walk on the 2-dimensional integer lattice begins at the origin. At each step, the walker moves one unit either left, right, or up, each with probability 1/3. (No downward steps ever.)

A walk is a success if it reaches the point (1, 1).

What is the probability of success?