There is a triangle with one ant on each of its 3 vertices. Each ant moves at…
2024
There is a triangle with one ant on each of its 3 vertices. Each ant moves at the same speed and independently picks one of the two directions along the edges of the triangle at random to start moving. What is the probability that two ants collide (meet while approaching each other along the same edge)?
- A.
1/3
- B.
2/3
- C.
3/8
- D.
3/4
Attempted by 1 students.
Show answer & explanation
Correct answer: D
Concept: When several actors each make one of two equally likely, independent choices, the total number of equally likely outcomes is 2n (multiplication principle). If an event is easiest to describe by what does NOT happen, compute the complement first and use P(event) = 1 − P(complement).
Application:
Each of the 3 ants independently picks one of 2 directions to move along an edge of the triangle, so the total number of equally likely direction-combinations is 23 = 8.
A collision happens whenever two ants end up moving toward each other along the same edge.
The only combinations that produce NO collision are the two extreme cases where every ant travels in the same rotational direction around the triangle (all clockwise, or all anticlockwise).
So exactly 2 of the 8 combinations are collision-free, giving P(no collision) = 2/8 = 1/4.
By the complement rule, P(collision) = 1 − 1/4 = 3/4.
Cross-check: Writing out all 8 direction-combinations directly (each ant's choice labelled L for one direction and R for the other) confirms only two of them, LLL and RRR, are collision-free; the remaining six combinations each have at least one pair of ants approaching each other, so 6/8 = 3/4, the same result.