Three ants are sitting at the three corners of an equilateral triangle. Each…

Concept: When several independent objects each choose one of a fixed number of directions/outcomes at random, the total sample space is the product of each individual object's choice count, and any qualifying event's probability equals (number of favourable outcomes) divided by (total outcomes).

Application: Each of the 3 ants independently picks one of 2 directions to move along its triangle edge - clockwise or anti-clockwise.

1. Total sample space = 2 x 2 x 2 = 23 = 8 equally likely direction combinations.

2. Two ants sharing an edge collide unless every ant moves in the same rotational sense around the triangle.

3. If even one ant breaks from that common direction, it moves toward one of its two neighbours, causing a collision on that edge.

4. So exactly 2 of the 8 combinations avoid collision: all three ants clockwise, or all three ants anti-clockwise.

5. Probability of no collision = 2/8 = 1/4 = 0.25.

Cross-check: Computing it as a product of independent probabilities gives the same result: P(all clockwise) = 0.5 x 0.5 x 0.5 = 0.125, and P(all anti-clockwise) = 0.5 x 0.5 x 0.5 = 0.125. These are mutually exclusive events, so P(no collision) = 0.125 + 0.125 = 0.25, matching the counting approach.