Zebras on Triangle - No Collision
Three zebras are chilling in the desert. Suddenly a lion attacks! Each zebra is sitting on a corner of an equilateral triangle (sides of equal length). Simultaneously, each zebra randomly picks a direction (it can only choose to run towards one of the two adjacent corners along an edge) and starts moving along the outline of the triangle to that chosen adjacent vertex.
What is the probability that none of the zebras collide?
Related Concepts
Hint
Let's break this down step-by-step:
- Choices for one zebra: Each zebra is at a corner. It can run along one edge or the other. So, how many choices does one zebra have? (Answer: 2 choices – e.g., "left" or "right" from its perspective, which translates to clockwise or counter-clockwise around the triangle).
- Total choices for all zebras:
- Zebra 1 has its number of choices.
- Zebra 2 has its number of choices (and its choice doesn't depend on Zebra 1's choice).
- Zebra 3 has its number of choices.
- When do they not collide? Think about what has to happen for them to miss each other. If one zebra goes clockwise and another goes counter-clockwise, will they meet? (Yes, either at a corner or along an edge). So, for no collision, they all need to be "in sync." What does that mean in terms of their direction around the triangle?
- Count the "no collision" scenarios: How many specific combinations of choices lead to no one bumping into each other? (Consider: what if they all go clockwise? What if they all go counter-clockwise?)
- Calculate the probability: Probability = (Number of "no collision" ways) / (Total number of ways they can run).
Explanation: Zebras on Triangle - No Collision
Imagine three zebras: Alex, Marty, and Gloria. They are each standing at a different corner of a triangular patch of grass. Suddenly, a lion roars! Each zebra instantly panics and decides to run to one of the other two corners along the edge of the grass patch. They all pick a direction randomly and at the same time.
The Question: What's the chance that none of them bump into each other?
Let's figure out all the possibilities:
- What can Alex do? Alex is at one corner. There are two edges leading away from Alex's corner. So, Alex has 2 choices of where to run. (Let's call them "left" or "right" from Alex's point of view, or more generally, clockwise or counter-clockwise around the triangle).
- What can Marty do? Marty also has 2 choices, independent of Alex.
- What can Gloria do? Gloria also has 2 choices, independent of the others.
Total Ways They Can Run (All Possible Outcomes): Since each zebra makes its own choice, and there are 3 zebras, the total number of different ways they could all run is: Zebra 1's choices × Zebra 2's choices × Zebra 3's choices = 2 × 2 × 2 = 8 possible scenarios. (For example, Alex goes left, Marty goes left, Gloria goes left. Or Alex goes left, Marty goes left, Gloria goes right. And so on, for 8 total combinations).
When Do They Not Collide? (Favorable Outcomes): They won't collide if they all move in a coordinated way, essentially following each other around the triangle.
- Scenario 1: All run Clockwise. Imagine Alex runs to Marty's original spot, Marty runs to Gloria's original spot, and Gloria runs to Alex's original spot. They are all moving in the same "circular" direction. No collisions! This is one specific way out of the 8.
- Scenario 2: All run Counter-Clockwise. Similarly, if Alex runs the other way, and Marty and Gloria also run in that same counter-clockwise direction, they will also miss each other. This is another specific way out of the 8.
In any other scenario, they will collide. For example, if Alex and Marty run towards each other along one edge, they collide. If Alex and Marty run towards Gloria's corner, they both arrive at the same corner and collide (or at least have a very awkward zebra traffic jam!).
So, there are only 2 ways for them to run without colliding.
Calculating the Chance (Probability): The probability of an event is: (Number of ways the event can happen) / (Total number of possible ways things can happen).
Probability of no collision = (Number of "no collision" ways) / (Total number of ways they can run)
P(No Collision) = 2 / 8
This fraction can be simplified:
P(No Collision) = 1/4
As a decimal, 1/4 is 0.25. As a percentage, it's 25%.
So, there's a 1/4 (or 25%) chance that none of the zebras will collide.
Let's denote the three zebras as Z1, Z2, and Z3, positioned at the vertices of an equilateral triangle. Each zebra independently and randomly chooses one of two directions to run along an edge towards an adjacent vertex. These directions can be thought of as clockwise (CW) or counter-clockwise (CCW) relative to the triangle's perimeter.
1. Determine the Total Number of Possible Outcomes (Sample Space)
Each zebra has 2 independent choices for its direction:
- Zebra 1: 2 choices (CW or CCW)
- Zebra 2: 2 choices (CW or CCW)
- Zebra 3: 2 choices (CW or CCW)
Total Possible Outcomes = 2 × 2 × 2 = 23 = 8
These 8 outcomes are all equally likely since each zebra's choice is random.
2. Identify the Favorable Outcomes (No Collision)
The zebras will not collide if and only if they all choose to move in the same rotational direction around the triangle. If their directions are mixed (e.g., some go CW and some go CCW), they will inevitably meet at a vertex or along an edge.
The two scenarios where no collision occurs are:
- All three zebras move in the Clockwise (CW) direction. (Z1 moves CW, Z2 moves CW, Z3 moves CW) - This is 1 specific outcome.
- All three zebras move in the Counter-Clockwise (CCW) direction. (Z1 moves CCW, Z2 moves CCW, Z3 moves CCW) - This is 1 specific outcome.
In any other combination of choices (e.g., CW-CW-CCW, CW-CCW-CW, etc.), at least two zebras will attempt to occupy the same edge segment heading towards each other, or arrive at the same vertex from different directions simultaneously, leading to a collision.
Thus, there are 2 favorable outcomes where no collision occurs.
3. Calculate the Probability
The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes:
P(No Collision) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Plugging in our numbers:
P(No Collision) = 2 / 8
This fraction simplifies to:
P(No Collision) = 1/4
As a decimal, this is 0.25, or 25%.
Key Insight: The crucial insight for this type of problem is recognizing the condition for no collision. On a closed loop like a triangle (or any polygon), if individuals at each vertex move to an adjacent vertex, they will only avoid collision if they all move in a coordinated rotational direction (either all clockwise or all counter-clockwise). Any deviation by even one individual from this coordinated movement will lead to a collision. This problem is a good example of how simple rules for individual agents can lead to interesting collective outcomes, and how basic probability principles can be used to analyze them.
Generalize: What would be the probability of no collision if there were N zebras on an N-sided regular polygon, each randomly choosing to move along an edge to an adjacent vertex? Does the pattern from the triangle hold?