Ants on a Triangle

Solution & Explanation

The problem of determining the probability that three ants on the vertices of an equilateral triangle will avoid colliding involves understanding the possible movements of each ant and the conditions under which they do not collide.

Problem Setup:

• Vertices: A, B, and C of an equilateral triangle.
• Ants: Positioned at each vertex.
• Movement: Each ant can move either clockwise (CW) or counter-clockwise (CCW) along the edges of the triangle.

Possible Outcomes:

Each ant has two choices of direction, leading to a total of:

• $2 \times 2 \times 2 = 2^3 = 8$ possible combinations of movements.

Non-Collision Scenarios:

The ants will avoid colliding if they all move in the same direction:

1. All move clockwise (CW): (CW, CW, CW)
2. All move counter-clockwise (CCW): (CCW, CCW, CCW)

These two scenarios ensure that the ants do not meet each other on any edge of the triangle.

Probability Calculation:

• Total favorable outcomes: 2 (all CW or all CCW)
• Total possible outcomes: 8

The probability that the ants will not collide is calculated by the ratio of favorable outcomes to the total possible outcomes:

$P(\text{no collision}) = \frac{\text{Number of non-colliding combinations}}{\text{Total number of combinations}} = \frac{2}{8} = \frac{1}{4} = 0.25$

Explanation of Independence:

• Each ant's choice of direction is independent of the others, meaning the decision of one ant does not affect the others. This is why we can multiply the probabilities of individual directions to find the total probability of a particular outcome.

Conclusion:

The probability that the ants will avoid colliding with one another, by all moving in the same direction around the triangle, is $0.25$ or $25\%$. This solution considers all possible movements and identifies the conditions for non-collision, leading to a straightforward calculation based on the symmetry of the problem.