一个游戏有两组图标，每组各有八个图标。每一轮从每组中随机选择一个图标进行显示，从第 1 组开始。每组的图标如下。 例如，如果从第 1 组中选择了一个 (三个相扣的圆圈) 图标，并从第 2 组中选择了一个 (帆船) 图标，则显示为 (三个相扣的圆圈)(帆船)。

1. Find the probability of getting the required icon from each set and multiply them.

Set 1 has $4$ three-interlocking-circles icons out of $8$ icons.

Set 2 has $2$ three-interlocking-circles icons out of $8$ icons.

$P(\text{three-interlocking-circles from Set 1 and Set 2})=\frac{4}{8}\times\frac{2}{8}=\frac{8}{64}=\frac{1}{8}$

1. Find the probability that no three-interlocking-circles icon is chosen from either set.

Set 1 has $4$ non-three-interlocking-circles icons out of $8$, and Set 2 has $6$ non-three-interlocking-circles icons out of $8$.

$P(\text{no three-interlocking-circles})=\frac{4}{8}\times\frac{6}{8}=\frac{3}{8}$

2. Use the complement to find the probability of at least one three-interlocking-circles icon.

$P(\text{at least one three-interlocking-circles})=1-P(\text{no three-interlocking-circles})$

$=1-\frac{3}{8}=\frac{5}{8}$

1. Write the first-stage probabilities for choosing each icon from Set 1.

Let
$C$ = three-interlocking-circles, $S$ = sailing-boat, $R$ = rocket, $G$ = graduation-cap.

Set 1 has $4C, 1S, 2R, 1G$, out of $8$ icons.

So the first branches are
$P(C)=\frac{4}{8},\quad P(S)=\frac{1}{8},\quad P(R)=\frac{2}{8},\quad P(G)=\frac{1}{8}.$

2. Find the second-stage probabilities after each possible Set 1 choice, removing the same icon type from Set 2.

Set 2 originally has $2C, 2S, 3R, 1G$.

If $C$ is chosen from Set 1, remove $2C$ from Set 2. Remaining: $2S,3R,1G$, total $6$.
$P(S\mid C)=\frac{2}{6},\quad P(R\mid C)=\frac{3}{6},\quad P(G\mid C)=\frac{1}{6}.$

If $S$ is chosen from Set 1, remove $2S$ from Set 2. Remaining: $2C,3R,1G$, total $6$.
$P(C\mid S)=\frac{2}{6},\quad P(R\mid S)=\frac{3}{6},\quad P(G\mid S)=\frac{1}{6}.$

If $R$ is chosen from Set 1, remove $3R$ from Set 2. Remaining: $2C,2S,1G$, total $5$.
$P(C\mid R)=\frac{2}{5},\quad P(S\mid R)=\frac{2}{5},\quad P(G\mid R)=\frac{1}{5}.$

If $G$ is chosen from Set 1, remove $1G$ from Set 2. Remaining: $2C,2S,3R$, total $7$.
$P(C\mid G)=\frac{2}{7},\quad P(S\mid G)=\frac{2}{7},\quad P(R\mid G)=\frac{3}{7}.$

3. Present the complete tree diagram structure with all possible non-matching displayed pairs.

The tree diagram should show:

$$\begin{array}{c|c} \text{Set 1 branch} & \text{Set 2 branches after removal} \\ \hline C\;\frac{4}{8} & S\;\frac{2}{6},\; R\;\frac{3}{6},\; G\;\frac{1}{6} \\ S\;\frac{1}{8} & C\;\frac{2}{6},\; R\;\frac{3}{6},\; G\;\frac{1}{6} \\ R\;\frac{2}{8} & C\;\frac{2}{5},\; S\;\frac{2}{5},\; G\;\frac{1}{5} \\ G\;\frac{1}{8} & C\;\frac{2}{7},\; S\;\frac{2}{7},\; R\;\frac{3}{7} \end{array}$$

No branch has the same icon in both positions.