Posterior probability after robot movement

Suppose robot moves in cells and there are 5 cells. Their dist. is as follows: | 1/9 | 1/3 | 1/3 | 1/9 | 1/9 |

The robot moves one cells towards right. And the world is cyclic. When it moves to the most right cell, it return back to the most left one.

And the posterior probability after one cell movement is as follows: | 1/9 | 1/9 | 1/3 | 1/3 | 1/9 |

The following diagram is a good illustration.

Can any guy tell me why the posterior probability shifts to the right one cell? Thanks in advance!

Think about the probability of the robot being in any cell A at Time t in terms of its probability of being or not being in cell A-1 at Time t-1:

Break event up into mutually exclusive joint events:

--> P(Robot loc @ T = A ) = P(Robot loc @ T=A, Robot loc @ T-1 = A-1) + P(Robot loc @ T=A, Robot loc @ T-1 <> A-1)

Use conditional probability to break those joint events up into independent events:

--> P(Robot loc @ T= A ) = P(Robot loc @ T=A | Robot loc @ T-1 = A-1) . P(Robot loc @ T-1 = A-1) + P(Robot loc @ T=A | Robot loc @ T-1 <> A-1) . P(Robot loc @ T-1 <> A-1)

And that allows us to use the fact that the robot is moving to the right (the event that the robot has moved to the right has prob 1, any other possibility has prob 0).

--> P(Robot loc @ T= A ) = 1 . P(Robot loc @ T-1 = A-1) + 0 . P(Robot loc @ T-1 <> A-1)

Simplify, and get the answer you wanted.

--> P(Robot loc @ T= A ) = P(Robot loc @ T-1 = A-1)