8 Athletes are standing on vertices of a cube. They can run in any direction. Find out the probability that there is no collision. All the athletes have same speed. Athletes can move only on edges. Athletes can be considered as dots.
I have thought the following:
Probability of no collisions + Probability of atleast 1 collision = 1 This covers the whole spectrum of possibilities.
Changing sides of P(atleast 1 collision).
P(No collisions) = 1 - P(atleast 1 collision)
Further expanding probability of at least one collision:
P(atleast 1 collision) = P(1 collision) + P(2 collisions) ..... P(12 collisions).
I am unable to proceed further.
Start by looking at some one athlete. It picks direction and runs. In order not to collide with the athlete who stood on the other side of the edge, that other athlete should have picked any other direction. Etc.
If you carefully proceed on the edges - they form a cycle. Any athlete is on some cycle. on the cube smallest cycle is of length $4$. Therefore the only possible cycle of athletes are $4$ and $8$ ($6$ is not possible, since no cycle of length $2$ - for remaining two athletes is possible)
The first option is two distinct loops of length $4$. Which means two quartet groups of athletes run in parallel planes (see the image below). How many such loops are possible? in each you can pick the direction independently (i.e. $4$ options) and there are $3$ pairs of planes. In total $12$ different ways (not to collide).
The second option, they all run in the same cycle (see image below). How many such cycles are possible on the cube? I've arbitrary picked a node on the cube and "started" the cycle with colored edges. First two edges uniquely identifies the cycle (try to continue the path after first two and you see you ought to pick very specific turns). Once you have picked a red edge - you have two options to pick the blue one. At the beginning there are $3$ options to set the red edge. And obviously the movement direction on the cycle (two of them). Totaling in another $12$ cycles.
You have $24$ ways to pick direction without collisions. And there is obviously $3^8 = 6561$ ways to pick any direction in total.
So, a probability of $0.0036$ to not collide.