Answer :
To determine which statements about Yuri's experiment are true, let's analyze the information step-by-step in detail.
1. Theoretical Probability of Spinning Any One of the Five Colors:
- Since the spinner has five congruent sections, the probability of landing on any one color for a single spin is:
[tex]\[ \text{Theoretical Probability} = \frac{1}{5} = 0.2 = 20\% \][/tex]
So, the first statement is true.
2. Experimental Probability of Spinning Blue:
- Yuri's results show that blue came up 1 time out of 10 spins.
- The experimental probability of spinning blue is:
[tex]\[ \text{Experimental Probability (Blue)} = \frac{\text{Number of Blue Spins}}{\text{Total Spins}} = \frac{1}{10} = 0.1 \][/tex]
- Comparing this with the statement [tex]\(\frac{1}{5}\)[/tex], which is 0.2, we see that 0.1 is not equal to 0.2.
Therefore, the second statement is false.
3. Theoretical Probability of Spinning Green Equals Experimental Probability of Spinning Green:
- The theoretical probability of landing on green is the same as any other color:
[tex]\[ \text{Theoretical Probability (Green)} = \frac{1}{5} = 0.2 \][/tex]
- Yuri's results show that green came up 2 times out of 10 spins:
[tex]\[ \text{Experimental Probability (Green)} = \frac{\text{Number of Green Spins}}{\text{Total Spins}} = \frac{2}{10} = 0.2 \][/tex]
- Since the theoretical and experimental probabilities are both 0.2, the third statement is true.
4. Experimental Probability of Spinning Yellow is Less Than Theoretical Probability of Spinning Yellow:
- The theoretical probability of landing on yellow is:
[tex]\[ \text{Theoretical Probability (Yellow)} = \frac{1}{5} = 0.2 \][/tex]
- Yuri's results show that yellow came up 3 times out of 10 spins:
[tex]\[ \text{Experimental Probability (Yellow)} = \frac{\text{Number of Yellow Spins}}{\text{Total Spins}} = \frac{3}{10} = 0.3 \][/tex]
- Comparing 0.3 to 0.2, we see that 0.3 is not less than 0.2.
Therefore, the fourth statement is false.
5. Law of Large Numbers and Spinning Orange:
- According to the law of large numbers, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
- If Yuri spins the spinner 600 more times, the experimental probability of spinning orange should approach the theoretical probability of [tex]\(0.2\)[/tex]:
[tex]\[ \text{Theoretical Probability (Orange)} = 0.2 \quad \text{because each section is congruent}. \][/tex]
Therefore, the fifth statement is true.
Given this analysis, the three correct statements are:
- The theoretical probability of spinning any one of the five colors is [tex]\(20\%\)[/tex].
- The theoretical probability of spinning green equals the experimental probability of spinning green.
- 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.
1. Theoretical Probability of Spinning Any One of the Five Colors:
- Since the spinner has five congruent sections, the probability of landing on any one color for a single spin is:
[tex]\[ \text{Theoretical Probability} = \frac{1}{5} = 0.2 = 20\% \][/tex]
So, the first statement is true.
2. Experimental Probability of Spinning Blue:
- Yuri's results show that blue came up 1 time out of 10 spins.
- The experimental probability of spinning blue is:
[tex]\[ \text{Experimental Probability (Blue)} = \frac{\text{Number of Blue Spins}}{\text{Total Spins}} = \frac{1}{10} = 0.1 \][/tex]
- Comparing this with the statement [tex]\(\frac{1}{5}\)[/tex], which is 0.2, we see that 0.1 is not equal to 0.2.
Therefore, the second statement is false.
3. Theoretical Probability of Spinning Green Equals Experimental Probability of Spinning Green:
- The theoretical probability of landing on green is the same as any other color:
[tex]\[ \text{Theoretical Probability (Green)} = \frac{1}{5} = 0.2 \][/tex]
- Yuri's results show that green came up 2 times out of 10 spins:
[tex]\[ \text{Experimental Probability (Green)} = \frac{\text{Number of Green Spins}}{\text{Total Spins}} = \frac{2}{10} = 0.2 \][/tex]
- Since the theoretical and experimental probabilities are both 0.2, the third statement is true.
4. Experimental Probability of Spinning Yellow is Less Than Theoretical Probability of Spinning Yellow:
- The theoretical probability of landing on yellow is:
[tex]\[ \text{Theoretical Probability (Yellow)} = \frac{1}{5} = 0.2 \][/tex]
- Yuri's results show that yellow came up 3 times out of 10 spins:
[tex]\[ \text{Experimental Probability (Yellow)} = \frac{\text{Number of Yellow Spins}}{\text{Total Spins}} = \frac{3}{10} = 0.3 \][/tex]
- Comparing 0.3 to 0.2, we see that 0.3 is not less than 0.2.
Therefore, the fourth statement is false.
5. Law of Large Numbers and Spinning Orange:
- According to the law of large numbers, as the number of trials increases, the experimental probability tends to get closer to the theoretical probability.
- If Yuri spins the spinner 600 more times, the experimental probability of spinning orange should approach the theoretical probability of [tex]\(0.2\)[/tex]:
[tex]\[ \text{Theoretical Probability (Orange)} = 0.2 \quad \text{because each section is congruent}. \][/tex]
Therefore, the fifth statement is true.
Given this analysis, the three correct statements are:
- The theoretical probability of spinning any one of the five colors is [tex]\(20\%\)[/tex].
- The theoretical probability of spinning green equals the experimental probability of spinning green.
- 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.
