There are three ants at the three corners of a regular
triangle. Each ant starts moving on a straight line toward
another, randomly chosen corner. What is the probability
that none of the ants collide?
25%. From the ant's perspective each can go left or right. There are 8 poss combinations - LLL, LLR, LRL, RLL, LRR, RLR, RRL, and RRR. You can also work that out by raising 2 (the number of choices) to the power of 3 (the number of choosers) = 2 x 2 x 2 = 8. In only 2 of these cases (LLL and RRR will there be no collisions, so 2 / 8 = 0.25
There are two directions to move for each ant
Thus total 2x2x2 =8 movements in different directions
out of these only two result in non - collisions of any of the ant - either all clockwise or all counter clockwise
Thus rest 6 result in at leaat 1 collission.
Thus probablility of non collision = 2/8=1/4=0.25
Reference : http://puzzles4you.blogspot.com/2008/03/three-ants.html
1 comment:
Notice that their can be only one collision. For the ants that collide share a common side, of which the end points are the starting points of these ants.
The other end is travelling along one of the other two sides and will reach each destination before encountering a other ant.
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