A spinner has five congruent sections, one each of blue, green, red, orange, and yellow. Yuri spins the spinner 10 times and records his results in the table.
| Color | Number |
| :----- | :----- |
| blue | 1 |
| green | 2 |
| red | 0 |
| orange | 4 |
| yellow | 3 |
Which statements are true about Yuri's experiment? Select three options.
A. The theoretical probability of spinning any one of the five colors is 20%.
B. The experimental probability of spinning blue is [tex]$\frac{1}{5}$[/tex].
C. The theoretical probability of spinning green is equal to the experimental probability of spinning green.
D. The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow.
E. 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.
Answer (1)
The theoretical probability of each color is 5 1 = 20% .
The experimental probabilities are: blue 10 1 , green 5 1 , red 0 , orange 10 4 , yellow 10 3 .
The true statements are: theoretical probability is 20%; experimental and theoretical probability of green are equal; with more spins, the experimental probability of orange will approach the theoretical probability.
The true statements are: theoretical probability is 20%; experimental and theoretical probability of green are equal; with more spins, the experimental probability of orange will approach the theoretical probability.
Explanation
Analyze the problem Let's analyze the problem. We have a spinner with five equal sections, each with a different color: blue, green, red, orange, and yellow. Yuri spins the spinner 10 times and records the results. We need to determine which of the given statements are true based on the theoretical and experimental probabilities.
Calculate theoretical probability First, let's calculate the theoretical probability of landing on each color. Since there are five congruent sections, the theoretical probability of landing on any one color is 5 1 , which is equal to 0.2 or 20% .
Calculate experimental probabilities Now, let's calculate the experimental probabilities based on Yuri's results:
Blue: Yuri spun blue 1 time out of 10, so the experimental probability is 10 1 .
Green: Yuri spun green 2 times out of 10, so the experimental probability is 10 2 = 5 1 .
Red: Yuri spun red 0 times out of 10, so the experimental probability is 10 0 = 0 .
Orange: Yuri spun orange 4 times out of 10, so the experimental probability is 10 4 = 5 2 .
Yellow: Yuri spun yellow 3 times out of 10, so the experimental probability is 10 3 .
Evaluate the statements Now, let's evaluate the given statements:
The theoretical probability of spinning any one of the five colors is 20%. This is true, as we calculated the theoretical probability to be 5 1 = 20% .
The experimental probability of spinning blue is 5 1 . This is false. The experimental probability of spinning blue is 10 1 .
The theoretical probability of spinning green is equal to the experimental probability of spinning green. This is true. The theoretical probability is 5 1 , and the experimental probability is 10 2 = 5 1 .
The experimental probability of spinning yellow is less than the theoretical probability of spinning yellow. The experimental probability of spinning yellow is 10 3 = 0.3 . The theoretical probability is 5 1 = 0.2 . Since 0.2"> 0.3 > 0.2 , this statement is false.
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 is true. As the number of trials increases, the experimental probability tends to converge to the theoretical probability.
Identify true statements Therefore, the true statements are:
The theoretical probability of spinning any one of the five colors is 20%.
The theoretical probability of spinning green is equal to 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.
Examples
Understanding theoretical and experimental probabilities helps in many real-world scenarios. For example, in quality control, manufacturers use experimental data to ensure their products meet theoretical standards. Similarly, in sports, analyzing a player's past performance (experimental probability) can help predict their future success, compared against the average performance (theoretical probability). These concepts are also fundamental in fields like finance, where understanding the likelihood of different outcomes is crucial for making informed decisions.
