Question

gerald is conducting an experiment with 3 possible outcomes. kasey is conducting an experiment with 20 possible outcomes. which statement best relates the possible outcomes to the number of trials needed to produce results similar to their predicted outcomes?
gerald will need to conduct fewer trials because experimental and theoretical results in experiments with small numbers of possible outcomes are the same.
gerald will need to conduct more trials because he needs to make up for the difference in the number of possible outcomes.
gerald will need to conduct fewer trials than kasey because there are fewer possible outcomes in his experiment.
gerald will need to conduct more trials than kasey because there are fewer possible outcomes in his experiment.

Explanation:

Step1: Understand the relationship between number of outcomes and trials

In probability, when there are more possible outcomes in an experiment, more trials are generally needed to get results that are close to the theoretical probability. Gerald has an experiment with 20 possible outcomes and Kasey has one with 3 possible outcomes.

Step2: Analyze each option

• Option 1: "Gerald will need to conduct fewer trials because experimental and theoretical results in experiments with small numbers of possible outcomes are the same." - This is incorrect. Just because there are fewer outcomes doesn't mean results are the same. And Gerald has more outcomes, so he can't conduct fewer trials.
• Option 2: "Gerald will need to conduct more trials because he needs to make up for the difference in the number of possible outcomes." - This is not a correct reasoning. The number of trials is related to getting accurate results based on the number of outcomes, not "making up" for something.
• Option 3: "Gerald will need to conduct fewer trials than Kasey because there are fewer possible outcomes in his experiment." - Incorrect. Gerald has more outcomes (20 vs 3), so he can't conduct fewer trials.
• Option 4: "Gerald will need to conduct more trials than Kasey because there are fewer possible outcomes in his experiment." - Correct. Since Kasey has fewer outcomes (3), fewer trials can give a relatively accurate result. But Gerald has 20 outcomes. To get results similar to the predicted (theoretical) outcomes (which assume all 20 outcomes are equally likely in a fair experiment), he needs more trials to account for the larger number of possible results.

Answer:

Gerald will need to conduct more trials than Kasey because there are fewer possible outcomes in his experiment.