To solve this problem, let's first understand the difference between experimental probability and theoretical probability.
1. Theoretical Probability:
- The theoretical probability of rolling a specific number on a 6-sided die (such as rolling a 2) is given by:
[tex]\[
P(\text{Rolling a 2}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{6}
\][/tex]
2. Experimental Probability:
- The experimental probability is based on the actual results of the trials. Jamie rolled the die 30 times and found that the experimental probability of rolling a 2 is [tex]\(\frac{1}{15}\)[/tex]. This means that out of 30 rolls, the number 2 appeared:
[tex]\[
\text{Number of times 2 appeared} = \left(\frac{1}{15}\right) \times 30 = 2 \text{ times}
\][/tex]
Given the discrepancy between the experimental probability [tex]\(\frac{1}{15}\)[/tex] and the theoretical probability [tex]\(\frac{1}{6}\)[/tex], Jamie needs to consider how he could adjust his process to make the experimental probability more accurately reflect the theoretical probability.
One fundamental principle in probability is the Law of Large Numbers, which states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. Therefore:
- If Jamie increases the number of trials, the experimental results should converge closer to the theoretical probability [tex]\(\frac{1}{6}\)[/tex].
Considering the options given:
1. He can increase the number of trials: This would help his experimental results better align with the theoretical probability over time.
2. He can decrease the number of trials: This would actually lead to more variation and less accuracy, making the experimental probability less reliable.
3. He can increase the number of sides on the die: This would change the theoretical probability entirely, making it no longer [tex]\(\frac{1}{6}\)[/tex].
4. He can decrease the number of sides on the die: This also changes the nature of the problem and the theoretical probability.
Therefore, the most appropriate action Jamie could take is to increase the number of trials. This approach will help ensure his experimental results come closer to the theoretical probability of rolling a 2, which is [tex]\(\frac{1}{6}\)[/tex].
The correct choice is:
- He can increase the number of trials.
