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Each ant can run in clockwise or in an Anti-clockwise direction. So there can be 8 possible ways in which they can start running. They will not collide only if all 3 run in either clockwise or in an Anti-clockwise direction.
So the required probability is 2/8 = 1/4.
An after thought - this holds good only if the ants run with the same speed otherwise the ants will collide.
Binominal formula will help here. Ants have 2 options; to take on one or the other side of each vertex. The prob being equal-1/2 and i get (1/2+1/2) ^3=1/8+3/8+3/8+1/8. We have the case where all 3 start to go in clockwise direction or all 3 start in counterclockwise direction. Which is 1/8+1/8=1/4
The ans is 1/4