Question

you are playing a board game and your playing piece begins the game at start. you roll a single number cube numbered 1 to 6 to find out how many spaces you can move. what is the theoretical probability of landing on a question mark space on your first roll.
a ( \frac{1}{6} )

b ( \frac{1}{4} )

c ( \frac{1}{3} )

d ( \frac{1}{2} )

Explanation:

Step1: Count total spaces

From START, the first 6 spaces (since die is 1 - 6) are: 1 (START is 0, first move 1: green), 2: green, 3: yellow (star), 4: blue (?), 5: blue (?), 6: yellow (star)? Wait, no, looking at the board: Let's list the spaces from START (position 0) when moving 1 - 6:

- Move 1: green (space 1)
- Move 2: green (space 2)
- Move 3: yellow (star, space 3)
- Move 4: blue (?, space 4)
- Move 5: blue (?, space 5)
- Move 6: yellow (star, space 6)? Wait, no, the left path: START, then green, green, yellow (star), blue (?), blue (?), yellow (star), then the curve. Wait, actually, the question mark spaces in the first 6 moves: Let's count the number of question mark spaces in the first 6 positions (since rolling 1 - 6, so moving 1 to 6 spaces from START).

Looking at the board:

- Space 1 (move 1): green
- Space 2 (move 2): green
- Space 3 (move 3): yellow (star)
- Space 4 (move 4): blue (?)
- Space 5 (move 5): blue (?)
- Space 6 (move 6): yellow (star)? No, wait the left side: START, then green (1), green (2), yellow (star, 3), blue (?, 4), blue (?, 5), yellow (star, 6)? No, the top left: after yellow (star, 3), there are two blue (?) spaces (4 and 5), then yellow (star, 6)? Wait, no, the diagram: START, then two green, one yellow (star), two blue (?), one yellow (star) – that's 6 spaces? Wait, no, let's count the question mark spaces in the first 6 moves. Wait, the first 6 spaces (moves 1 - 6) have how many? Let's see:

Wait, the board:

- Move 1: green (space 1)
- Move 2: green (space 2)
- Move 3: yellow (star, space 3)
- Move 4: blue (?, space 4)
- Move 5: blue (?, space 5)
- Move 6: yellow (star, space 6)? No, that's 6 spaces. Wait, no, the left path: START, then green (1), green (2), yellow (star, 3), blue (?, 4), blue (?, 5), yellow (star, 6) – so in moves 1 - 6, the question mark spaces are at move 4 and 5? Wait, no, maybe I miscounted. Wait, the original board: looking at the image, the left side (from START) has:

START → green (1) → green (2) → yellow (star, 3) → blue (?, 4) → blue (?, 5) → yellow (star, 6) → then the curve. Wait, no, the top part: after yellow (star, 6), there's a blue (?)? No, the image shows:

Left column (from START up):

- START (bottom)
- green (1)
- green (2)
- yellow (star, 3)
- blue (?, 4)
- blue (?, 5)
- yellow (star, 6)
- then the curve to the right: blue (?, 7), yellow (star, 8), etc. But the first roll is 1 - 6, so moves 1 - 6: spaces 1 - 6.

Wait, no, maybe the first 6 spaces (moves 1 - 6) have 2 question marks? No, wait the answer options: 1/3 is 2/6? Wait, no, 1/3 is 2/6? Wait, 3 question marks? Wait, maybe I miscounted. Let's look again:

The board:

From START (bottom left), moving up:

1. green (move 1)
2. green (move 2)
3. yellow (star, move 3)
4. blue (?, move 4)
5. blue (?, move 5)
6. yellow (star, move 6)
7. blue (?, move 7) – but first roll is 1 - 6, so move 7 is beyond. Wait, no, maybe the first 6 spaces (moves 1 - 6) have 2 question marks? But 2/6 = 1/3? Wait, 2 question marks in 6 spaces? No, 2/6 is 1/3. Wait, maybe the correct count is: in the first 6 moves (spaces 1 - 6), how many question marks?

Wait, the image: the left path (from START) has:

- Space 1: green
- Space 2: green
- Space 3: yellow (star)
- Space 4: blue (?)
- Space 5: blue (?)
- Space 6: yellow (star)
- Then space 7: blue (?) (top left)
- Space 8: yellow (star)
- Space 9: blue (?)
- Etc. But the first roll is 1 - 6, so spaces 1 - 6. So question marks at 4 and 5: 2 spaces. Wait, but 2/6 = 1/3. So probability is 2/6 = 1/3. Hence option C.

Step2: Calculate probability

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Favorable outcomes (question mark spaces in first 6 moves): 2? Wait, no, wait maybe I miscounted. Wait, looking at the board again: the first 6 spaces (moves 1 - 6) – let's count the question marks:

- Move 4:?
- Move 5:?
- Move 7:? (but move 7 is beyond first roll). Wait, no, maybe the first 6 spaces include 3 question marks? Wait, the image shows:

Top left: three blue (?) spaces? Let's see:

From START, moving up:

1. green
2. green
3. yellow (star)
4. blue (?)
5. blue (?)
6. yellow (star)
7. blue (?)
8. yellow (star)
9. blue (?)

Wait, no, the original image: the left curve (top) has three blue (?) spaces? Let's count the blue (?) spaces in the first 6 moves (moves 1 - 6):

Wait, move 1: green (1)

move 2: green (2)

move 3: yellow (star) (3)

move 4: blue (?) (4)

move 5: blue (?) (5)

move 6: yellow (star) (6)

move 7: blue (?) (7) – but first roll is 1 - 6, so move 7 is not included. So in moves 1 - 6, there are 2 question marks? But 2/6 = 1/3. So probability is 2/6 = 1/3, which is option C.

Answer:

C. \( \frac{1}{3} \)