题目
(a) Find the number of different arrangements of the 8 letters in the word KANGAROO in which the two As are together and the two Os are not together.
A fair 8-sided dice has faces labelled K, A, N, G, A, R, O, O. The dice is rolled repeatedly.
(b) Find the probability that fewer than 6 rolls of this dice are required to obtain an A.
(c) Find the probability that the second A is obtained on the 6th roll of the dice.
题目中文翻译
(a) 将单词 KANGAROO 中的 8 个字母排列，要求两个 A 相邻，两个 O 不相邻。求不同排列的个数。
一枚均匀的八面骰子的各面分别标有 K、A、N、G、A、R、O、O。反复掷这枚骰子。
(b) 求掷出 A 所需的次数少于 6 次的概率。
(c) 求第 6 次掷骰子时才第二次掷出 A 的概率。
原卷通用作答要求
Answer all questions. You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。
解答
(a)
解法一
思路
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将AA作为一个块，先算全部排列，再减去OO也相连的排列。
答题过程
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Treat AA as a single block. Together with K, N, G, R, O and O, there are seven objects with two identical Os, giving arrangements.
If OO is also a block, there are six distinct objects, giving arrangements. Hence
解法二
思路
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也可以先排AA块和K、N、G、R五个对象，再从六个空隙中选两个，每处放一个O。
答题过程
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Arrange AA, K, N, G and R in ways, creating six gaps including the end gaps. Place the identical Os in two distinct gaps so they are not together. Thus
(b)
解法一
思路
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骰子有两个A面，成功概率为四分之一。少于6次即前5次至少出现一个A，用补集计算。
答题过程
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On each roll,
Therefore,
解法二
思路
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另一条路线按首次A出现于第1至第5次分类，分别乘之前没有A的概率再求和。
答题过程
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Let be the number of rolls to the first A. Then
(c)
解法一
思路
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第6次是第二个A，前5次必须恰有一个A，第6次再成功。组合数安排前5次中的成功位置。