Answer :
Let's analyze the question step by step.
We have the following recorded results of Yuri's spins:
- Blue: 1
- Green: 2
- Red: 0
- Orange: 4
- Yellow: 3
- Total spins: 10
### Theoretical Probabilities:
Because the spinner has five congruent sections, each section should theoretically have an equal probability of being landed on. Therefore, the theoretical probability for each color is:
[tex]\[ \text{Theoretical Probability} = \frac{1}{5} = 0.2 = 20\% \][/tex]
### Experimental Probabilities:
The experimental probability is the ratio of the number of times the spinner landed on a particular color to the total number of spins.
- Experimental probability of blue:
[tex]\[ \text{Probability(blue)} = \frac{\text{Number of Blue Spins}}{\text{Total Spins}} = \frac{1}{10} = 0.1 \][/tex]
- Experimental probability of green:
[tex]\[ \text{Probability(green)} = \frac{\text{Number of Green Spins}}{\text{Total Spins}} = \frac{2}{10} = 0.2 \][/tex]
- Experimental probability of red:
[tex]\[ \text{Probability(red)} = \frac{\text{Number of Red Spins}}{\text{Total Spins}} = \frac{0}{10} = 0 \][/tex]
- Experimental probability of orange:
[tex]\[ \text{Probability(orange)} = \frac{\text{Number of Orange Spins}}{\text{Total Spins}} = \frac{4}{10} = 0.4 \][/tex]
- Experimental probability of yellow:
[tex]\[ \text{Probability(yellow)} = \frac{\text{Number of Yellow Spins}}{\text{Total Spins}} = \frac{3}{10} = 0.3 \][/tex]
### Analysis of Statements:
1. The theoretical probability of spinning any one of the five colors is [tex]$20 \%$[/tex].
- This statement is true since the theoretical probability for each color is [tex]\(20\%\)[/tex].
2. The experimental probability of spinning blue is [tex]\( \frac{1}{5} \)[/tex].
- This statement is false. The actual experimental probability of spinning blue is [tex]\(0.1 = 10\%\)[/tex], not [tex]\(\frac{1}{5} = 20\%\)[/tex].
3. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
- This statement is true as both the theoretical and experimental probabilities of green are [tex]\(0.2\)[/tex].
4. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
- This statement is false. The experimental probability of spinning yellow is [tex]\(0.3\)[/tex], which is greater than the theoretical probability of [tex]\(0.2\)[/tex].
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.
- This statement is true based on the law of large numbers, which states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
### 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.
We have the following recorded results of Yuri's spins:
- Blue: 1
- Green: 2
- Red: 0
- Orange: 4
- Yellow: 3
- Total spins: 10
### Theoretical Probabilities:
Because the spinner has five congruent sections, each section should theoretically have an equal probability of being landed on. Therefore, the theoretical probability for each color is:
[tex]\[ \text{Theoretical Probability} = \frac{1}{5} = 0.2 = 20\% \][/tex]
### Experimental Probabilities:
The experimental probability is the ratio of the number of times the spinner landed on a particular color to the total number of spins.
- Experimental probability of blue:
[tex]\[ \text{Probability(blue)} = \frac{\text{Number of Blue Spins}}{\text{Total Spins}} = \frac{1}{10} = 0.1 \][/tex]
- Experimental probability of green:
[tex]\[ \text{Probability(green)} = \frac{\text{Number of Green Spins}}{\text{Total Spins}} = \frac{2}{10} = 0.2 \][/tex]
- Experimental probability of red:
[tex]\[ \text{Probability(red)} = \frac{\text{Number of Red Spins}}{\text{Total Spins}} = \frac{0}{10} = 0 \][/tex]
- Experimental probability of orange:
[tex]\[ \text{Probability(orange)} = \frac{\text{Number of Orange Spins}}{\text{Total Spins}} = \frac{4}{10} = 0.4 \][/tex]
- Experimental probability of yellow:
[tex]\[ \text{Probability(yellow)} = \frac{\text{Number of Yellow Spins}}{\text{Total Spins}} = \frac{3}{10} = 0.3 \][/tex]
### Analysis of Statements:
1. The theoretical probability of spinning any one of the five colors is [tex]$20 \%$[/tex].
- This statement is true since the theoretical probability for each color is [tex]\(20\%\)[/tex].
2. The experimental probability of spinning blue is [tex]\( \frac{1}{5} \)[/tex].
- This statement is false. The actual experimental probability of spinning blue is [tex]\(0.1 = 10\%\)[/tex], not [tex]\(\frac{1}{5} = 20\%\)[/tex].
3. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
- This statement is true as both the theoretical and experimental probabilities of green are [tex]\(0.2\)[/tex].
4. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
- This statement is false. The experimental probability of spinning yellow is [tex]\(0.3\)[/tex], which is greater than the theoretical probability of [tex]\(0.2\)[/tex].
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.
- This statement is true based on the law of large numbers, which states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
### 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.
