Answer :
To determine which statements about Yuri's spinner experiment are true, let's examine each one in detail using the provided data and probabilities.
### Given Data:
- Total spins: 10
- Results:
- Blue: 1
- Green: 2
- Red: 0
- Orange: 4
- Yellow: 3
### Theoretical Probability:
Given that there are five congruent sections, the theoretical probability of landing on any one color is:
[tex]\[ P(\text{any color}) = \frac{1}{5} = 0.20 \text{ or } 20\% \][/tex]
### Experimental Probabilities:
We calculate the experimental probability for each color based on Yuri's recorded results.
1. Blue:
[tex]\[ P(\text{blue}) = \frac{\text{count of blue spins}}{\text{total spins}} = \frac{1}{10} = 0.10 \][/tex]
2. Green:
[tex]\[ P(\text{green}) = \frac{2}{10} = 0.20 \][/tex]
3. Red:
[tex]\[ P(\text{red}) = \frac{0}{10} = 0 \][/tex]
4. Orange:
[tex]\[ P(\text{orange}) = \frac{4}{10} = 0.40 \][/tex]
5. Yellow:
[tex]\[ P(\text{yellow}) = \frac{3}{10} = 0.30 \][/tex]
### Statements Analysis:
1. The theoretical probability of spinning any one of the five colors is [tex]\(20\%\)[/tex].
- As calculated, the theoretical probability is indeed [tex]\(20\%\)[/tex].
- This statement is True.
2. The experimental probability of spinning blue is [tex]\(\frac{1}{5}\)[/tex].
- The experimental probability of spinning blue is [tex]\(0.10\)[/tex] or [tex]\(10\%\)[/tex].
- [tex]\(\frac{1}{5}\)[/tex] is equal to 0.20 or 20%.
- This statement is False.
3. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
- Both the theoretical probability and the experimental probability for green are [tex]\(0.20\)[/tex] or [tex]\(20\%\)[/tex].
- This statement is True.
4. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
- The experimental probability of spinning yellow is [tex]\(0.30\)[/tex] or [tex]\(30\%\)[/tex].
- The theoretical probability of spinning yellow is [tex]\(0.20\)[/tex] or [tex]\(20\%\)[/tex].
- [tex]\(0.30\)[/tex] is not less than [tex]\(0.20\)[/tex]; it's actually greater.
- This statement is False.
5. If Yuri spins the spinner 600 more times and records results, the experimental probability of spinning orange will get closer to the theoretical probability of spinning orange.
- As the number of trials increases, the law of large numbers states that the experimental probabilities will converge to the theoretical probabilities.
- This statement is True.
### Conclusion:
The three true statements are:
1. The theoretical probability of spinning any one of the five colors is [tex]\(20\%\)[/tex].
2. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
3. If Yuri spins the spinner 600 more times and records results, the experimental probability of spinning orange will get closer to the theoretical probability of spinning orange.
These are the three statements that are true about Yuri's experiment.
### Given Data:
- Total spins: 10
- Results:
- Blue: 1
- Green: 2
- Red: 0
- Orange: 4
- Yellow: 3
### Theoretical Probability:
Given that there are five congruent sections, the theoretical probability of landing on any one color is:
[tex]\[ P(\text{any color}) = \frac{1}{5} = 0.20 \text{ or } 20\% \][/tex]
### Experimental Probabilities:
We calculate the experimental probability for each color based on Yuri's recorded results.
1. Blue:
[tex]\[ P(\text{blue}) = \frac{\text{count of blue spins}}{\text{total spins}} = \frac{1}{10} = 0.10 \][/tex]
2. Green:
[tex]\[ P(\text{green}) = \frac{2}{10} = 0.20 \][/tex]
3. Red:
[tex]\[ P(\text{red}) = \frac{0}{10} = 0 \][/tex]
4. Orange:
[tex]\[ P(\text{orange}) = \frac{4}{10} = 0.40 \][/tex]
5. Yellow:
[tex]\[ P(\text{yellow}) = \frac{3}{10} = 0.30 \][/tex]
### Statements Analysis:
1. The theoretical probability of spinning any one of the five colors is [tex]\(20\%\)[/tex].
- As calculated, the theoretical probability is indeed [tex]\(20\%\)[/tex].
- This statement is True.
2. The experimental probability of spinning blue is [tex]\(\frac{1}{5}\)[/tex].
- The experimental probability of spinning blue is [tex]\(0.10\)[/tex] or [tex]\(10\%\)[/tex].
- [tex]\(\frac{1}{5}\)[/tex] is equal to 0.20 or 20%.
- This statement is False.
3. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
- Both the theoretical probability and the experimental probability for green are [tex]\(0.20\)[/tex] or [tex]\(20\%\)[/tex].
- This statement is True.
4. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
- The experimental probability of spinning yellow is [tex]\(0.30\)[/tex] or [tex]\(30\%\)[/tex].
- The theoretical probability of spinning yellow is [tex]\(0.20\)[/tex] or [tex]\(20\%\)[/tex].
- [tex]\(0.30\)[/tex] is not less than [tex]\(0.20\)[/tex]; it's actually greater.
- This statement is False.
5. If Yuri spins the spinner 600 more times and records results, the experimental probability of spinning orange will get closer to the theoretical probability of spinning orange.
- As the number of trials increases, the law of large numbers states that the experimental probabilities will converge to the theoretical probabilities.
- This statement is True.
### Conclusion:
The three true statements are:
1. The theoretical probability of spinning any one of the five colors is [tex]\(20\%\)[/tex].
2. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
3. If Yuri spins the spinner 600 more times and records results, the experimental probability of spinning orange will get closer to the theoretical probability of spinning orange.
These are the three statements that are true about Yuri's experiment.
