33. The arrow on the spinner shown is equally likely to point in any direction after it is spun. Which of the following is a valid method for finding the theoretical probability that the arrow will point to the red sector when it is spun?
Answer: A
Use a protractor to determine the measure, in degrees, of the central angle of the red sector. Then divide the measure by 360 degrees.
This method accurately calculates the theoretical probability of the arrow pointing to the red sector by using the proportion of the angle representing the red sector relative to the total angle of the spinner.
A) Use a protractor to determine the measure, in degrees, of the central angle of the red sector. Then divide the measure by 360 degrees.
This is the correct method for finding the theoretical probability. By measuring the central angle of the red sector and dividing it by the total angle of the spinner (360 degrees), you obtain the proportion of the spinner that is red, directly reflecting its theoretical probability.
B) Use a protractor to determine the measure, in degrees, of the central angle of the red sector. Then divide the measure by the circumference of the spinner.
This option is incorrect because it uses the circumference as a divisor instead of the total angle. The probability should be based on angular measurement, not linear measurement, making this method invalid for calculating theoretical probability.
C) Spin the arrow 200 × and record the number of × it points to the red sector. Then find the relative frequency with which the arrow points to the red sector.
While this method can provide an empirical estimate of the probability through relative frequency, it does not determine the theoretical probability. The question specifically asks for a theoretical approach, making this option unsuitable.
D) Spin the arrow 20 × and record the number of × it points to the red sector. Repeat the process an additional 9 ×, then find the mean relative frequency with which the arrow points to the red sector for the 10 trials.
Similar to option C, this method aims to find the empirical probability based on trials, rather than the theoretical probability. Although it may yield useful data, it does not directly calculate the theoretical probability as required by the question.
Conclusion
The method in option A is the only valid approach for determining the theoretical probability, as it utilizes the central angle of the red sector in relation to the total angle of the spinner. All other options either misapply the concept of probability or focus on empirical methods, which do not satisfy the requirement for a theoretical calculation.