题目
Bag contains 8 red marbles and 3 blue marbles. Bag contains 4 red marbles and 1 blue marble.
A marble is chosen at random from bag . If the marble chosen is red, it is discarded. If the marble chosen is blue, it is placed in bag .
A marble is then chosen at random from bag . If the marble chosen is red, it is discarded. If the marble chosen is blue, it is placed in bag .
A marble is now chosen at random from bag .
(a) Complete the tree diagram below by entering all the remaining outcomes and probabilities.
(b) Find the probability that all three marbles chosen are the same colour.
题目中文翻译
袋 中有 8 颗红色弹珠和 3 颗蓝色弹珠。袋 中有 4 颗红色弹珠和 1 颗蓝色弹珠。
从袋 中随机取出一颗弹珠。若取出的是红色弹珠，就将其弃去；若取出的是蓝色弹珠，就将其放入袋 。
接着从袋 中随机取出一颗弹珠。若取出的是红色弹珠，就将其弃去；若取出的是蓝色弹珠，就将其放入袋 。
现在从袋 中随机取出一颗弹珠。
(a) 填入其余所有结果及概率，补全下方的树状图。
(b) 求取出的三颗弹珠颜色全部相同的概率。
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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依次更新每个袋中的红蓝数量。红珠被丢弃，蓝珠则转移到另一袋，因此最后一次从A袋取珠的概率取决于前两次的完整路径。
答题过程
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The completed branches are:
| First choice from A | Second choice from B | Final choice from A |
|---|---|---|
| Red, | Red, | Red ; Blue |
| Red, | Blue, | Red ; Blue |
| Blue, | Red, | Red ; Blue |
| Blue, | Blue, | Red ; Blue |
For example, after Red then Blue, A has lost one red marble and gained a blue marble, leaving 7 red and 4 blue.
(b)
解法一
思路
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三珠同色只能是红红红或蓝蓝蓝。沿树图分别相乘，再相加。