Ebere 有一枚不均匀的硬币，掷出正面的概率为 $\frac14$。她还有两个装有彩色圆片的袋子。袋 $A$ 中有 5 个红色圆片和 3 个蓝色圆片；袋 $B$ 中有 4 个红色圆片和 5 个绿色圆片。

Ebere 掷这枚硬币。如果掷出正面，她从袋 $A$ 中随机逐个取出两个圆片，不放回。如果掷出反面，她从袋 $B$ 中随机逐个取出两个圆片，不放回。

(a) 画出标注完整的树状图表示这些信息。

(b) 求 Ebere 取到的两个圆片颜色相同的概率。

(c) 已知取到的圆片中至少有一个是红色，求 Ebere 掷硬币时掷出正面的概率。

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。

第一层表示硬币结果和所选袋子；第二、三层表示两次取圆片的颜色。不放回意味着第二次总数减少1，同色圆片数也随第一次结果改变。

First branch into head, $H$, with probability $\frac14$, and tail, $T$, with probability $\frac34$.

• After $H$, use bag $A$: the first disc is R with probability $\frac58$ or B with probability $\frac38$. After R, the next probabilities are $\frac47$ for R and $\frac37$ for B; after B, they are $\frac57$ for R and $\frac27$ for B.
• After $T$, use bag $B$: the first disc is R with probability $\frac49$ or G with probability $\frac59$. After R, the next probabilities are $\frac38$ for R and $\frac58$ for G; after G, they are $\frac48$ for R and $\frac48$ for G.

The fully labelled tree is:

两次同色的路径有A袋的红红、蓝蓝，以及B袋的红红、绿绿。每条路径乘分支概率，然后相加。

设条件事件为至少一片红色。分子是选A袋且至少一红；分母可用1减去两片蓝和两片绿。注意两片红也属于条件事件。

Let $E$ be the event that at least one disc is red. A head selects bag $A$, so

The only outcomes with no red disc are BB from bag $A$ and GG from bag $B$. Hence

Therefore,