Ants on Triangle - No Collision
Three ants are placed, one on each vertex of a triangle. Simultaneously, each ant randomly chooses a direction (clockwise or counterclockwise) and starts moving along an edge of the triangle to an adjacent vertex. What is the probability that none of the ants collide?
Related Concepts
Hint
- How many choices does each ant have for its direction?
- Since the ants choose independently, how many total combinations of choices are there for all three ants? This is your total sample space.
- For the ants not to collide, what must be true about their chosen directions relative to each other? (Think about them all moving "in sync").
- How many ways can this "no collision" scenario happen?
- The probability is the number of "no collision" scenarios divided by the total number of scenarios.
Explanation: Ants on a Triangle - No Collision
Imagine three ants, Ant A, Ant B, and Ant C, each on a corner of a triangle.
Each ant will randomly decide to walk either clockwise or counter-clockwise to the next corner. They all move at the same time.
We want to find the chance that they don't bump into each other.
- Each Ant's Choice: Ant A has 2 choices (clockwise or counter-clockwise). Ant B also has 2 choices. Ant C also has 2 choices.
- Total Possible Scenarios: To find all possible ways they can move, we multiply their choices: 2 × 2 × 2 = 8 total scenarios.
- When DON'T they collide? They won't collide if they all move in the same direction.
- Scenario 1: All three ants move clockwise. (This is 1 way)
- Scenario 2: All three ants move counter-clockwise. (This is 1 way)
- The Chance: Out of 8 total possible scenarios, 2 of them result in no collision. So the probability is 2 out of 8.
Let the three ants be A, B, and C, positioned on the vertices of a triangle. Each ant independently chooses to move either clockwise (CW) or counter-clockwise (CCW) to an adjacent vertex.
1. Determine the Total Number of Possible Outcomes (Sample Space)
Each of the three ants has 2 independent choices for its direction of movement:
- Ant 1: 2 choices (CW or CCW)
- Ant 2: 2 choices (CW or CCW)
- Ant 3: 2 choices (CW or CCW)
Using the fundamental counting principle, the total number of possible outcomes in the sample space is:
Total Outcomes = 2 × 2 × 2 = 2³ = 8
These 8 outcomes are assumed to be equally likely since each ant chooses its direction randomly.
2. Identify the Favorable Outcomes (No Collision)
The ants will not collide if and only if they all choose to move in the same direction. If even one ant moves in a different direction from the others (relative to the overall flow), a collision is inevitable on a triangle (two ants would meet at a vertex or along an edge).
The scenarios where no collision occurs are:
- All ants move clockwise (CW):
(Ant 1: CW, Ant 2: CW, Ant 3: CW) - This is 1 specific outcome. - All ants move counter-clockwise (CCW):
(Ant 1: CCW, Ant 2: CCW, Ant 3: CCW) - This is 1 specific outcome.
Therefore, there are 2 favorable outcomes where the ants do not collide.
3. Calculate the Probability
The probability of an event (E) is calculated as:
P(E) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
In this case, E is the event that the ants do not collide.
P(No Collision) = 2 / 8
This fraction can be simplified:
P(No Collision) = 1/4
As a decimal, this is:
P(No Collision) = 0.25 or 25%
Why other scenarios lead to collision: Consider any other combination of choices. For example, if Ant 1 moves CW, Ant 2 moves CCW, and Ant 3 moves CW. Ant 1 and Ant 2 are moving towards each other on one edge, leading to a collision. On a triangle, any mix of CW and CCW movements will result in at least two ants meeting at a vertex or along an edge. The only way to avoid this is if they all maintain their relative positions by moving in the same rotational direction.
Puzzle Extension: What if there were 'N' ants on an N-sided regular polygon (N-gon)? What would be the probability of no collision then? Does the pattern hold?