On the probability of forming polygons from a broken stick

斐波那契数与随机长度棍棒形成多边形的概率

Fibonacci numbers and the probability of polygon formation using random length sticks

通过两种互补证明，得出随机长度棍棒无法形成p+1边多边形的概率可表示为涉及p步斐波那契数的项倒数乘积。

A Simple Multiple Integral Solution to the Broken Stick Problem

Regard the closed interval $[0,1]$ as a stick. Partition $[0,1]$ into $n+1$ different intervals $I_1, \ \dots \ , I_{n+1},$ where $n \geq 2,$ which represent smaller sticks. The classical Broken Stick problem asks to find the probability that the lengths of these smaller sticks can be the side lengths of a polygon with $n+1$ sides. We will show that this probability is $1-\frac{n+1}{2^{n}}$ by using multiple integration.

抽取木棍与斐波那契阶乘

Pick-up Sticks and the Fibonacci Factorial

本文研究了随机采样n根棍子后，无法构成三角形的概率，证明其与斐波那契数的乘积有关，并扩展至多边形问题。

Triangles, Fractales and Spaghetti

There is well-known problem of geometric probability which can be quote as the Broken Spaghetti Problem. It addresses the following question: A stick of spaghetti breaks into three parts and all points of the stick have the same probability to be a breaking point. What is the probability that the three sticks, putting together, form a triangle? In these notes, we describe a hidden geometric pattern behind the symmetric version of this problem, namely a fractal that parametrizes the sample space of this problem. Using that fractal, we address the question about the probability to obtain a $δ$-e…

概率与歧义

Probability and Ambiguity

本文通过分析 Gardner (2001) 中的断棍、两个男孩、三个囚犯问题及 Hamming (1991) 的钝角随机三角形问题，探讨概率计算中问题陈述的歧义性及其对概率函数和随机性理解的影响。