臺灣博碩士論文加值系統

摘要
假設有A1, A , , An 2  共n 個人進行剪刀石頭布的賽局，賽局依下列規則進行：
1 A 與2 A 先猜拳，勝利者再與3 A 猜拳，其中的勝利者再與4 A 猜拳其餘依此類推；
至於1 A 與2 A 猜拳後的敗方，必須等3 A n ,, A 皆比賽後才可上場比賽，其餘依此
類推。我們規定只要有人先連贏k 場則賽局結束，並稱此人贏得比賽。我們有興
趣的是各個玩家贏得比賽的機率及結束賽局所需場次的機率分配。本論文將用解
聯立方程式的方法推導出各個玩家贏得比賽的機率及利用機率生成函數求得結
束賽局所需場次的機率分配。

Abstract
Assume there are , 1 A n A , , A 2  n people participate in a rock-paper-scissor game,
and the game has to follow several rules: first, 1 A play the game with 2 A , the winner
of the game then play with 3 A , and so on; as for the losing side, they have to wait
until 3 A n ,, A completed the game. If the person successfully pass each set of the
game n times in a row, he will ultimately win the entire game. What we are
interested here is the probability of winning for each participant and the number of
games might take in order to finish the game. This thesis will use the probability
generating function in order to get the probability of winning for each participant and
the average number of games might take in order to finish the game

目次
第一章 Waldegrave 問題
第一節 研究背景及動機 2
第二節 n=2,3 時參賽者贏得比賽的機率 3
第三節 n ≥ 4時各參賽者贏得比賽的機率 8
第四節 n=2,3 時所需對局次數之機率分配 18
第五節 n ≥ 4時所需對局次數之機率分配 22
第二章 Waldegrave 問題的推廣
第一節 n=4,k=3 時各個參賽者獲勝的機率 24
第二節 n=5,k=3 時各個參賽者獲勝的機率 27
第三節 一般n , k的情形 30
參考文獻

A History of Probability and Statistics and their applications before 1750. Hald, Ander. John Wiley &; Sons, Inc., New York, 1990
Problems and Snapshots from the World of Probability. Blom, Gunnar; Holst, Lars; Sandell, Dennis. Springer-Verlag, New York, 1994.