Question

jonas is conducting an experiment using a 10 - sided die. he determines that the theoretical probability of rolling a 3 is \\(\frac{1}{10}\\). he rolls the die 20 times. four of those rolls result in a 3. which adjustment can jonas make to his experiment so the theoretical and experimental probabilities are likely to be closer? he can decrease the sample space. he can increase the sample space. he can decrease the number of trials. he can increase the number of trials.

Explanation:

Step1: Recall the Law of Large Numbers

The Law of Large Numbers states that as the number of trials (or experiments) increases, the experimental probability (the ratio of the number of favorable outcomes to the total number of trials) approaches the theoretical probability.

Step2: Analyze the effect of changing sample space

The sample space of a 10 - sided die is fixed (the set of possible outcomes \(\{1,2,\cdots,10\}\)). Changing the sample space (either increasing or decreasing it) would change the theoretical probability itself, not make the experimental probability closer to the original theoretical probability. For example, if we change the sample space, we are essentially changing the die (e.g., using an 11 - sided die or a 9 - sided die), which would give a new theoretical probability, not help the experimental probability of the 10 - sided die's roll of 3 to get closer to \(\frac{1}{10}\).

Step3: Analyze the effect of changing the number of trials

• Decreasing the number of trials: If we decrease the number of trials, the experimental probability is more likely to deviate from the theoretical probability because there is less data to average out the randomness. For example, with only 1 or 2 trials, we could get very extreme results (like rolling a 3 twice in 2 trials, giving an experimental probability of 1, or never rolling a 3, giving a probability of 0).
• Increasing the number of trials: When we increase the number of trials, the random fluctuations in the experimental results tend to average out. So, the experimental probability (based on the results of the die rolls) will get closer to the theoretical probability \(\frac{1}{10}\).

Answer:

He can increase the number of trials.