Answer :
Let's analyze the given information and statements step by step to determine their validity based on the given results.
1. The theoretical probability of spinning any one of the five colors is [tex]$20 \%$[/tex].
The spinner has five congruent sections, each representing a different color. Therefore, the probability of landing on any one specific color in a single spin is [tex]\(\frac{1}{5}\)[/tex], which equates to 20%.
- This statement is true.
2. The experimental probability of spinning blue is [tex]\(\frac{1}{5}\)[/tex].
The experimental probability of an event is calculated by dividing the number of times the event occurred by the total number of trials. Yuri spun the spinner 10 times and landed on blue 1 time. Thus, the experimental probability of spinning blue is:
[tex]\[ \frac{\text{Number of blue outcomes}}{\text{Total number of spins}} = \frac{1}{10} \][/tex]
- [tex]\(\frac{1}{10}\)[/tex] is not equal to [tex]\(\frac{1}{5}\)[/tex].
- This statement is false.
3. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
The theoretical probability of landing on green is the same as any other color, which is 20% or [tex]\(\frac{1}{5}\)[/tex].
The experimental probability of spinning green is calculated by:
[tex]\[ \frac{\text{Number of green outcomes}}{\text{Total number of spins}} = \frac{2}{10} = \frac{1}{5} \][/tex]
Since [tex]\(\frac{2}{10}\)[/tex] simplifies to [tex]\(\frac{1}{5}\)[/tex], the experimental probability of green aligns with the theoretical probability.
- This statement is true.
4. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
The theoretical probability of spinning yellow is 20% or [tex]\(\frac{1}{5}\)[/tex].
The experimental probability of spinning yellow is calculated by:
[tex]\[ \frac{\text{Number of yellow outcomes}}{\text{Total number of spins}} = \frac{3}{10} \][/tex]
[tex]\(\frac{3}{10}\)[/tex] is equivalent to 30%, which is more than 20%.
- This statement is false.
5. If Yuri spins the spinner 600 more times and records the 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 probability will tend to converge to the theoretical probability. Thus, if Yuri spins the spinner many more times, the experimental probability of landing on orange will likely become closer to the theoretical probability of 20%.
- This statement is true.
Based on the above analysis, the three true statements are:
1. The theoretical probability of spinning any one of the five colors is 20%.
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.
1. The theoretical probability of spinning any one of the five colors is [tex]$20 \%$[/tex].
The spinner has five congruent sections, each representing a different color. Therefore, the probability of landing on any one specific color in a single spin is [tex]\(\frac{1}{5}\)[/tex], which equates to 20%.
- This statement is true.
2. The experimental probability of spinning blue is [tex]\(\frac{1}{5}\)[/tex].
The experimental probability of an event is calculated by dividing the number of times the event occurred by the total number of trials. Yuri spun the spinner 10 times and landed on blue 1 time. Thus, the experimental probability of spinning blue is:
[tex]\[ \frac{\text{Number of blue outcomes}}{\text{Total number of spins}} = \frac{1}{10} \][/tex]
- [tex]\(\frac{1}{10}\)[/tex] is not equal to [tex]\(\frac{1}{5}\)[/tex].
- This statement is false.
3. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
The theoretical probability of landing on green is the same as any other color, which is 20% or [tex]\(\frac{1}{5}\)[/tex].
The experimental probability of spinning green is calculated by:
[tex]\[ \frac{\text{Number of green outcomes}}{\text{Total number of spins}} = \frac{2}{10} = \frac{1}{5} \][/tex]
Since [tex]\(\frac{2}{10}\)[/tex] simplifies to [tex]\(\frac{1}{5}\)[/tex], the experimental probability of green aligns with the theoretical probability.
- This statement is true.
4. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
The theoretical probability of spinning yellow is 20% or [tex]\(\frac{1}{5}\)[/tex].
The experimental probability of spinning yellow is calculated by:
[tex]\[ \frac{\text{Number of yellow outcomes}}{\text{Total number of spins}} = \frac{3}{10} \][/tex]
[tex]\(\frac{3}{10}\)[/tex] is equivalent to 30%, which is more than 20%.
- This statement is false.
5. If Yuri spins the spinner 600 more times and records the 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 probability will tend to converge to the theoretical probability. Thus, if Yuri spins the spinner many more times, the experimental probability of landing on orange will likely become closer to the theoretical probability of 20%.
- This statement is true.
Based on the above analysis, the three true statements are:
1. The theoretical probability of spinning any one of the five colors is 20%.
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.
