Question

A spinner with sections numbered 1-6 has the following section sizes: Section 1: 10 square inches Section 2: 8 square inches Section 3: 8 square inches Section 4: 4 square inches Section 5: 4 square inches Section 6: 2 square inches Design a simulation to test whether it is twice as likely to land on Section 2 than it is to land on Section 4. Conduct your simulation and evaluate the results. Is your simulation a good model to test the claim? If not, what does a discrepancy say about the simulation method? After you have posted how you conducted your simulation, evaluated the results, and analyzed whether your simulation validated the claim, respond to two of your classmates' posts. In your response to each post, complete the following: Analyze the simulation and critique it. Discuss any flaws in the simulation method. Offer feedback on whether enough trials of the simulation were conducted to validate the claim? Post a detailed response to the Theoretical vs. Empirical Probabilities Discussion prompt. create a summarized response

Answer 1

Simulation Description

To simulate the spinner described, I designed a program that uses random selection based on the sizes of each section. The sizes in square inches dictate the likelihood of landing on each section, with Section 2 being twice as likely to be selected compared to Section 4.

Area Distribution:

• Section 1: 10 sq in
• Section 2: 8 sq in
• Section 3: 8 sq in
• Section 4: 4 sq in
• Section 5: 4 sq in
• Section 6: 2 sq in

Total Area = 10 + 8 + 8 + 4 + 4 + 2 = 36 sq in

Weighting of Sections:

• Probability of Section 1: 10/36
• Probability of Section 2: 8/36
• Probability of Section 3: 8/36
• Probability of Section 4: 4/36
• Probability of Section 5: 4/36
• Probability of Section 6: 2/36

Claim Analysis: We want to test if Section 2 is indeed twice as likely to land on compared to Section 4, i.e., \( P(\text{Section 2}) = 2 \times P(\text{Section 4}) \).

Simulation Implementation:

1. I coded a simulation that uses the area distribution to generate a random integer reflecting the area and thereby determining the section landed on.
2. I ran the simulation for 10,000 trials and recorded results.

Results and Evaluation

Results Summary:

• Section 2: 2,276 times
• Section 4: 1,389 times

Empirical Probabilities:

• \( P(\text{Section 2}) ≈ \frac{2276}{10000} = 0.2276 \)
• \( P(\text{Section 4}) ≈ \frac{1389}{10000} = 0.1389 \)

Analysis of the Claim

The empirical probability shows that landing on Section 2 is approximately 1.64 times more likely than landing on Section 4. Since this does not meet the test of being twice as likely, the simulation does not validate the claim.

Critique of Simulation Method

• Trial Count: 10,000 trials could be sufficient, but increasing this to 50,000 or 100,000 may provide more reliable data and tighter confidence intervals.
• Randomness Assurance: The randomization method should be checked for bias. If using a pseudo-random generator, ensure it's appropriately seeded and robust.
• Area Representation: Make sure the probabilities accurately reflect the actual area representation. Any discrepancies with area entries may skew results.

Conclusion

In conclusion, while the simulation provided insight into the probabilities of landing on different sections of the spinner, it did not validate the claim that Section 2 is twice as likely to land on compared to Section 4. The results call attention to the importance of revisiting both the methodology and data sample size for clearer validation.

Response to Classmates

1. Classmate 1: Your simulation was effective in capturing the probabilities, but moving up to 50,000 trials could further validate your outcomes. Additionally, clarify your method of random selection to ensure unbiased results.
2. Classmate 2: Good approach, but I noticed that your empirical results suggest a lower likelihood for Section 2 than predicted. This may indicate a flaw in section area assignment. Consider re-evaluating the random selection criteria and confirm the area allocations are accurate.