Chance of Creating a Triangle from a Divided Stick

Solution & Explanation

The problem at hand is a classic probability puzzle often referred to as the "Broken Stick Problem." The main goal is to determine the probability that three segments, formed by breaking a stick at two random points, can form a triangle.

Problem Breakdown

• Initial Setup:
  • You have a stick of unit length (1 unit).
  • This stick is broken at two random points along its length, creating three segments.
• Objective:
  • Determine the probability that these three segments can form a triangle.

Triangle Inequality Theorem

To form a triangle from three segments, the triangle inequality theorem must hold:

• The sum of the lengths of any two segments must be greater than the length of the third segment.

Mathematical Representation

• Let the two break points be represented by random variables $X$ and $Y$ such that $0 < X < Y < 1$.
• This results in three segments:
  • Segment 1: $X$
  • Segment 2: $Y - X$
  • Segment 3: $1 - Y$

Applying the Triangle Inequality

1. Condition 1:

   • $X + (Y - X) > 1 - Y$
   • Simplifies to: $Y > 0.5$

2. Condition 2:

   • $X + (1 - Y) > Y - X$
   • Simplifies to: $X < 0.5$

3. Condition 3:

   • $(Y - X) + (1 - Y) > X$
   • This condition is always true because $0 < X < Y < 1$.

Geometric Interpretation

• The conditions $Y > 0.5$ and $X < 0.5$ define a triangular region in the unit square $0 < X < Y < 1$.
• The triangle is formed by the intersection of the lines $Y = 0.5$ and $X = 0.5$ within the unit square.
• This triangular region occupies one-fourth ($\frac{1}{4}$) of the total area of the unit square.

Conclusion

• Probability:
  • The probability that the three segments can form a triangle is $\frac{1}{4}$ or 0.25.

Simulation

To verify this probability, a simulation can be performed in Python:

import numpy as np

# Number of simulations
N = 1000000

# Generate random break points
X = np.random.uniform(0, 1, N)
Y = np.random.uniform(0, 1, N)

# Ensure X < Y
X, Y = np.minimum(X, Y), np.maximum(X, Y)

# Compute the number of successful trials
count = np.sum((X < 0.5) & (Y > 0.5))

# Calculate the probability
probability = count / N
print("Estimated Probability:", probability)

The simulation should yield a probability close to 0.25, affirming the theoretical solution.