## March 8, 2008

### Three Ants

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?

Anonymous said...

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

Secret SquÃ¯rrel

Neeraj said...

There are two directions tomove 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

Anonymous said...

Reverse the way the riddle is asked

Let’s start by reversing the way the riddle is asked and find the probability that there will not be a collision instead. When will this happen?

Well, what if all of the ants are walking in the same direction? Then there will never be a collision between any of the ants because they are all walking in the same direction. And, the only time they will be walking in the same direction is if they are all walking either counter-clockwise around the triangle, or clockwise around the triangle. So, there are only two scenarios in which a collision will not happen between the ants.
3 Ants on a Triangle Riddle Solution – the probability

Now that we know that there are only two scenarios where the ants will not collide, we have to ask ourselves how many different ways are there for the ants to move on the sides of the triangle? Well, each ant can move in 2 different directions. Because there are 3 ants, this means that there are 23 (which equals eight) possible ways that the ants can move. And since we already know that there are only 2 ways in which the ants can avoid collision entirely, this means that there are 6 scenarios where the ants will collide. And 6 out of 8 possible scenarios, means that the probability of collision is 6/8, which equals 3/4 or .75. Thus, the probability of the ants colliding is .75.