A problem I solved yesterday during my 10-minute subway ride.

Given a line segment of unit length, we put two points into it randomly so that it is divided into three segments. What is the probability that these three segments can form a triangle?

Let the two points be $x$ and $y$, and assume them to be uniformly distributed on [0:1]. If the three segments can form a triangle, it must satisfy the triangle inequality, i.e. no one side is longer than 0.5. Assume $x>y$, then we must have $% $ and $% $. The probability of such is

Similarly for the case $y>x$, we must have $% $ and $% $, and the probability is also $\frac{1}{8}$. So the total probability is $\frac{1}{4}$.

To verify the result, below is the Python code:

import random

def triangle():
(x,y) = (random.random(), random.random())
return 1 if max(min(x,y), abs(x-y), max(x,y)) < 0.5 else 0

def main():
N = 1000000
print "%f\n" % (sum(triangle() for i in xrange(N))/float(N))

if __name__ == '__main__':
main()


It prints 0.25