Triangle Formation from Random Stick Breaks

Solution & Explanation

The problem of determining the probability of forming a triangle from three segments obtained by breaking a stick at two random points can be understood through the lens of the triangle inequality theorem. This theorem states that, for any three segments to form a triangle, the sum of the lengths of any two segments must be greater than the length of the third segment.

Problem Setup

• Consider a stick of unit length (1).
• Break the stick at two random points, resulting in three segments.
• Determine the probability that these segments can form a triangle.

Understanding the Triangle Inequality

For three segments, say of lengths $a$, $b$, and $c$ (where $a + b + c = 1$), to form a triangle, they must satisfy:

1. $a + b > c$
2. $b + c > a$
3. $c + a > b$

Given the total length is 1, these inequalities can be rewritten as:

• $a + b > 0.5$
• $b + c > 0.5$
• $c + a > 0.5$

The condition implies that no single segment should be greater than 0.5.

Geometric Probability Approach

1. Random Breaks:

   • Imagine selecting two random points on the stick, $x$ and $y$, such that $0 < x < y < 1$.
   • The segments are $x$, $y - x$, and $1 - y$.

2. Triangle Formation Condition:

   • Each segment must be less than 0.5 to satisfy the triangle inequality.

3. Visualizing the Solution:

   • Consider a unit square with coordinates $(x, y)$ such that $0 < x < y < 1$.
   • The condition $x < 0.5$ and $y - x < 0.5$ and $1 - y < 0.5$ defines a region within this square.

4. Calculating the Probability:

   • The feasible region can be visualized as a triangle within the square, bounded by the lines $y = x + 0.5$, $y = 0.5$, and $x = 0.5$.
   • The area of this triangle is $\frac{1}{4}$ of the total unit square.

Conclusion

The probability that three segments formed by breaking a stick at two random points can form a triangle is $\frac{1}{4}$, or 25%. This result is derived by considering the geometric constraints imposed by the triangle inequality and visualizing the feasible region within a unit square.