Equilateral triangle T_{0} with side length 3 is on a plane. Given triangle T_{n} on the plane, triangle T_{n+1} is constructed on the plane by translating T_{n} by 1 unit, in one of six directions parallel to one of the sides of T_{n}. The direction is chosen uniformly at random.
Let a be the least integer such that at most one point on the plane is in or on all of T_{0}, T_{1}, T_{2}, \ldots, T_{a}. It can be shown that a exists with probability 1. Find the probability that a is even.