There are 3 ants at 3 corners of a triangle, they randomly start moving towards another corner. What is the probability that they don't collide?
Now, if I just simply look at the problem, I see that the three ants can simply go clockwise, or counterclockwise together. On the third possibility one ant could go clockwise, another counter clockwise, and the third ant wouldn't matter which way it went. This would be the collision making the answer 2/3, but this is wrong.
So lets take another look at it:
Lets name the ants A, B, C respectively, and to make it easier let 0 represent the ant goes clockwise, and 1 represent the ant goes counter-clockwise. We would end up with these possibilities:
0,0,0 = No Collision
0,0,1 = Collision
0,1,0 = Collision
0,1,1 = Collision
1,0,0 = Collision
1,0,1 = Collision
1,1,0 = Collision
1,1,1 = No Collision
This makes the correct answer of 2/8 = 1/4
Another way of looking at this is via the combinatorics way:
Each of the 3 ants has 2 possibilities, making it 2C1 for each ant. This comes out to 2*2*2 = 8 possibilities, now knowing that there are only 2 good outcomes this makes it 2/8 = 1/4.