To determine which experimental design will create probabilities that vary the most from their theoretical probabilities, we need to consider the law of large numbers. The law of large numbers states that as the number of trials increases, the experimental probability will get closer to the theoretical probability.
Let's break down each experimental design:
A. Roll a 6-sided die 12 times and count the number of 3's that occur.
- Since this involves only 12 trials, the probability results here are likely to vary more from the theoretical probability compared to a larger number of trials. Small sample sizes typically yield high variability.
B. Roll a 6-sided die 500 times and record the number of prime numbered outcomes that occur.
- Rolling the die 500 times represents a large number of trials, hence the results will tend to converge to the theoretical probability with less variability.
C. Roll a 6-sided die 200 times and record the number of even numbered outcomes that occur.
- Rolling the die 200 times is a significant number of trials, and the results will generally converge to the theoretical probability with relatively less variability.
D. Roll a 6-sided die 100 times and record the number of 5's that occur.
- This involves 100 trials. While not as high as 500 or 200, this number of trials will still provide less variability and more convergence towards theoretical probability compared to a very low number like 12.
From this analysis, rolling the die only 12 times (option A) is the smallest number of trials among the given options. As such, this design will have the most variability from the theoretical probabilities due to the small sample size.
Therefore, the correct answer is:
A. Roll a 6-sided die 12 times and count the number of 3's that occur.
