题目
Sam and Tom are playing a game which involves a bag containing 5 white discs and 3 red discs. They take turns to remove one disc from the bag at random. Discs that are removed are not replaced into the bag. The game ends as soon as one player has removed two red discs from the bag. That player wins the game.
Sam removes the first disc.
(a) Find the probability that Tom removes a red disc on his first turn.
(b) Find the probability that Tom wins the game on his second turn.
(c) Find the probability that Sam removes a red disc on his first turn given that Tom wins the game on his second turn.
题目中文翻译
Sam 和 Tom 玩一个游戏，使用的袋子中装有 5 个白色圆片和 3 个红色圆片。他们轮流从袋中随机取出一个圆片，取出的圆片不放回袋中。一旦某位玩家已从袋中取出两个红色圆片，游戏便结束，该玩家获胜。
Sam 先取出第一个圆片。
(a) 求 Tom 在他的第一轮取出红色圆片的概率。
(b) 求 Tom 在他的第二轮赢得游戏的概率。
(c) 已知 Tom 在他的第二轮赢得游戏，求 Sam 在他的第一轮取出红色圆片的概率。
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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Tom第一轮是总第2次，按Sam首次红或白两条路径求和。
答题过程
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(b)
解法一
思路
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Tom第二轮是第4次，需第2与第4次均红。Sam不能在第3次已赢，且总共只有3红，因此合法颜色次序恰三条。
答题过程
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Order the draws as Sam, Tom, Sam, Tom. Tom wins on his second turn on exactly the patterns RRWR, WRRR and WRWR. In each pattern Tom has two reds and Sam has at most one red before the final draw, so the game has not already ended.
Their probabilities are
Hence
(c)
解法一
思路
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在(b)合法路径中，Sam首次红只有RRWR，故用该路径除以(b)总概率。