SOLUTION: An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at

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Answer(since I originally answered the question in Latex, and since It's almost time to sleep, I will edit it tomorrow.)(The original post is here: https://artofproblemsolving.com/texer/yglroond)


Color the dots red and blue shown below.. Notice that whenever the ant moves, it moves from a red dot to a blue dot or a blue dot to a red dot. So since  is a red dot, it must move to a blue dot, then a red dot, then a blue dot, then a red dot, and end up on a blue dot. There are only four blue dots, and the ant is equally likely to end up on any one of these four, since the diagram is symmetric to a  rotation. The probability that the ant ends on  after five minutes is therefore .
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Let E denotes one step horizontally in (+) direction (East)
    W denotes one step horizontally in (-) direction (West)
    N denotes one step vertically   in (+) direction (North)
    S denotes one step vertically   in (-) direction (South).


We will denote each path as a sequence of letters E, W, N, S.


To get from A to B in 5 steps, the path must contain 5 letters; equal number of E and W letters;
                        in addition, the number of N letters must be 1 more than the number of S letters.


So, all good paths must fall into one of these DISJOINT categories


    (1) no E,W at all; there are only all distinguishable permutations of 3N and 2S.
                       the number of such arrangements is   =  = 10. 


    (2) one E compensated by one W, in any order; the rest are 2N and 1S in any arrangement.
                       the total number of such arrangements is  =  = 60.


    (3) two E compensated by two W, in any order; and only one N; no S
                       the total number of such artranjements is  =  = 30.


Thus, there are 10 + 60 + 30 = 100 good arrangements (paths leading from A to B).


The total number of all possible paths of the length 5 is  = 1024: any of 4 letters in each of 5 positions, independently.


So the probability is  P =  =  = 0.09766  (rounded).    ANSWER