Problem 6. A Piece of Broken Wood Forming a Triangle
. We randomly and unformly divide a piece of wood into two parts, and then randomly and uniformly divide the larger part into two parts. What is the probability that the three parts make a triangle?
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1668891586674188288
. The answer is equal to: \(2 \ln 2 - 1\).
In the first division, the size of the larger part will be equal to \(1/2 + x\), where the value of \(x\) is drawn randomly from the interval \([0,1/2]\). According to the triangle inequality, it is possible to make a triangle with the broken pieces if the size of the larger part of the second division is less than \(1/2\). The probability of this happening is equal to \((1/2 - x)/(1/2+x)\). So the answer to the question is equal to the following integral: \[\int_{0}^{\frac{1}{2}} x \frac{\frac{1}{2} - x}{\frac{1}{2} + x} d_x = 2\ln 2-1 \simeq 0.38629436112.\]
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