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Sagot :
Sure, let's break down the solution step-by-step to create an area model based on the given information:
1. Understand the Spin Outcomes:
- The wheel has 10 slots.
- Roll appears 6 times.
- Flip appears 4 times.
- Therefore, the probability of landing on Roll is [tex]\(\frac{6}{10} = 0.6\)[/tex] (or 60%).
- The probability of landing on Flip is [tex]\(\frac{4}{10} = 0.4\)[/tex] (or 40%).
2. Outcome for Roll:
- When the wheel lands on Roll, you roll a die.
- The die has 6 sides: 1 red, 2 blue, 3 green.
- So, the probability of rolling:
- Red is [tex]\(\frac{1}{6}\)[/tex].
- Blue is [tex]\(\frac{2}{6} = \frac{1}{3}\)[/tex].
- Green is [tex]\(\frac{3}{6} = \frac{1}{2}\)[/tex].
3. Outcome for Flip:
- When the wheel lands on Flip, you flip a coin.
- Assuming the coin is fair, the probabilities are:
- Heads [tex]\(\frac{1}{2}\)[/tex].
- Tails [tex]\(\frac{1}{2}\)[/tex].
Now, let's put this information into an area model.
### Area Model Structure:
1. Divide the space into two sections representing the outcomes of spinning the wheel: Roll and Flip.
2. Further subdivide these outcomes based on the resulting probabilities from die roll (for Roll) and coin flip (for Flip).
First Level:
- Roll (60%) and Flip (40%).
Second Level:
- For Roll:
- Red: [tex]\(0.6 \times \frac{1}{6} = 0.1\)[/tex]
- Blue: [tex]\(0.6 \times \frac{1}{3} = 0.2\)[/tex]
- Green: [tex]\(0.6 \times \frac{1}{2} = 0.3\)[/tex]
- For Flip:
- Heads: [tex]\(0.4 \times \frac{1}{2} = 0.2\)[/tex]
- Tails: [tex]\(0.4 \times \frac{1}{2} = 0.2\)[/tex]
### Constructing the Area Model:
1. Roll Outcome (Total 0.6 or 60%):
- Red (0.1 or 10%)
- Blue (0.2 or 20%)
- Green (0.3 or 30%)
2. Flip Outcome (Total 0.4 or 40%):
- Heads (0.2 or 20%)
- Tails (0.2 or 20%)
Visually it will look something like this:
```
---------------------------------------------
| Spin |
---------------------------------------------
| Roll (0.6) | Flip (0.4) |
---------------------------------------------
| Red | Blue | Green | | Heads | Tails |
| (0.1)| (0.2) | (0.3) | | (0.2) | (0.2) |
---------------------------------------------
```
In the area model:
- The width of the columns represents the probability of the spin result (Roll or Flip).
- The height within each column represents the probability of the secondary outcome (die roll or coin flip).
This completes our detailed, step-by-step construction of an area model based on the given situation.
1. Understand the Spin Outcomes:
- The wheel has 10 slots.
- Roll appears 6 times.
- Flip appears 4 times.
- Therefore, the probability of landing on Roll is [tex]\(\frac{6}{10} = 0.6\)[/tex] (or 60%).
- The probability of landing on Flip is [tex]\(\frac{4}{10} = 0.4\)[/tex] (or 40%).
2. Outcome for Roll:
- When the wheel lands on Roll, you roll a die.
- The die has 6 sides: 1 red, 2 blue, 3 green.
- So, the probability of rolling:
- Red is [tex]\(\frac{1}{6}\)[/tex].
- Blue is [tex]\(\frac{2}{6} = \frac{1}{3}\)[/tex].
- Green is [tex]\(\frac{3}{6} = \frac{1}{2}\)[/tex].
3. Outcome for Flip:
- When the wheel lands on Flip, you flip a coin.
- Assuming the coin is fair, the probabilities are:
- Heads [tex]\(\frac{1}{2}\)[/tex].
- Tails [tex]\(\frac{1}{2}\)[/tex].
Now, let's put this information into an area model.
### Area Model Structure:
1. Divide the space into two sections representing the outcomes of spinning the wheel: Roll and Flip.
2. Further subdivide these outcomes based on the resulting probabilities from die roll (for Roll) and coin flip (for Flip).
First Level:
- Roll (60%) and Flip (40%).
Second Level:
- For Roll:
- Red: [tex]\(0.6 \times \frac{1}{6} = 0.1\)[/tex]
- Blue: [tex]\(0.6 \times \frac{1}{3} = 0.2\)[/tex]
- Green: [tex]\(0.6 \times \frac{1}{2} = 0.3\)[/tex]
- For Flip:
- Heads: [tex]\(0.4 \times \frac{1}{2} = 0.2\)[/tex]
- Tails: [tex]\(0.4 \times \frac{1}{2} = 0.2\)[/tex]
### Constructing the Area Model:
1. Roll Outcome (Total 0.6 or 60%):
- Red (0.1 or 10%)
- Blue (0.2 or 20%)
- Green (0.3 or 30%)
2. Flip Outcome (Total 0.4 or 40%):
- Heads (0.2 or 20%)
- Tails (0.2 or 20%)
Visually it will look something like this:
```
---------------------------------------------
| Spin |
---------------------------------------------
| Roll (0.6) | Flip (0.4) |
---------------------------------------------
| Red | Blue | Green | | Heads | Tails |
| (0.1)| (0.2) | (0.3) | | (0.2) | (0.2) |
---------------------------------------------
```
In the area model:
- The width of the columns represents the probability of the spin result (Roll or Flip).
- The height within each column represents the probability of the secondary outcome (die roll or coin flip).
This completes our detailed, step-by-step construction of an area model based on the given situation.
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