Question

1. you and a friend invent a new board game. each player rolls a die to determine how many spaces to move, then uses a spinner to determine whether the player moves his/her own piece, the opponent’s piece, or both pieces. 8 marks the die is a custom die with faces labelled 1, 2, 3, 3, 4, 6. a) create a tree diagram to show the possible outcomes when the die is rolled and spinner spun in a turn.

Explanation:

Step1: Identify Die Outcomes

The die has faces: 1, 2, 3, 3, 4, 6. So the distinct die outcomes (with their counts, but for the tree, we list each face as a branch) are 1, 2, 3, 3, 4, 6. Wait, actually, for the tree, we consider each possible die roll result. So the first level of the tree (die roll) has branches: 1, 2, 3 (first 3), 3 (second 3), 4, 6. But since two 3s are distinct in terms of die faces (even though same number), but maybe we can group them as "3" with two branches? Wait, no—each face is a separate outcome. So the die has 6 faces: 1, 2, 3 (face 1), 3 (face 2), 4, 6. So six branches for the die.

Step2: Identify Spinner Outcomes

The spinner has three outcomes: move own piece (O), move opponent’s piece (Opp), move both (B). So for each die outcome, we have three spinner branches.

Step3: Construct the Tree

• Root (Start): Split into die outcomes: 1, 2, 3 (first), 3 (second), 4, 6.
• For each die outcome (e.g., 1): Split into spinner outcomes: O, Opp, B.
• Repeat for all die outcomes:
• Die = 1: branches to O, Opp, B
• Die = 2: branches to O, Opp, B
• Die = 3 (first): branches to O, Opp, B
• Die = 3 (second): branches to O, Opp, B
• Die = 4: branches to O, Opp, B
• Die = 6: branches to O, Opp, B

Visually, the tree diagram would have:

1. First level (die roll): 6 nodes (1, 2, 3, 3, 4, 6)
2. Second level (spinner): each die node connects to 3 nodes (O, Opp, B)

So the tree starts with the "Turn" as the root. Then first set of branches: 1, 2, 3, 3, 4, 6. Each of these branches then splits into three: O, Opp, B.

Answer:

To create the tree diagram:

Step 1: Die Roll Outcomes

The die has 6 faces: \( 1, 2, 3, 3, 4, 6 \). These form the first level (die roll) branches of the tree.

Step 2: Spinner Outcomes

The spinner has 3 outcomes: Move Own Piece (O), Move Opponent’s Piece (Opp), Move Both (B). These form the second level (spinner) branches for each die outcome.

Tree Diagram Structure (Textual Representation):

Start (Turn)
├── Die = 1
│   ├── Spinner: O (Own)
│   ├── Spinner: Opp (Opponent)
│   └── Spinner: B (Both)
├── Die = 2
│   ├── Spinner: O
│   ├── Spinner: Opp
│   └── Spinner: B
├── Die = 3 (Face 1)
│   ├── Spinner: O
│   ├── Spinner: Opp
│   └── Spinner: B
├── Die = 3 (Face 2)
│   ├── Spinner: O
│   ├── Spinner: Opp
│   └── Spinner: B
├── Die = 4
│   ├── Spinner: O
│   ├── Spinner: Opp
│   └── Spinner: B
└── Die = 6
    ├── Spinner: O
    ├── Spinner: Opp
    └── Spinner: B

This tree shows all possible combinations of die rolls (6 options) and spinner results (3 options), resulting in \( 6 \times 3 = 18 \) total outcomes (note: the two "3" die faces are distinct, so they create separate branches).