To find the smallest discrepancy between the experimental and expected probabilities based on the outcomes and their frequencies, we first need to calculate the total frequency and then determine the experimental probabilities for each outcome.
Step 1: Calculate the total frequency.
Total frequency = 10 (for 2) + 9 (for 4) + 6 (for 6) + 15 (for 8) + 13 (for 10) + 8 (for 12) = 61.
Step 2: Calculate the experimental probabilities.
Now we calculate the experimental probability for each outcome:
- P(2) = Frequency of 2 / Total frequency = 10 / 61
- P(4) = Frequency of 4 / Total frequency = 9 / 61
- P(6) = Frequency of 6 / Total frequency = 6 / 61
- P(8) = Frequency of 8 / Total frequency = 15 / 61
- P(10) = Frequency of 10 / Total frequency = 13 / 61
- P(12) = Frequency of 12 / Total frequency = 8 / 61
Step 3: Calculate the expected probabilities.
The expected probability for each outcome is based on uniform distribution since there are 6 outcomes. Therefore, the expected probability for each number is:
- E(2) = E(4) = E(6) = E(8) = E(10) = E(12) = 1/6 ≈ 0.1667
Step 4: Calculate the discrepancy for each outcome.
Discrepancy for each outcome = |Experimental Probability - Expected Probability|
- For 2: |(10/61) - (1/6)| = |0.1639 - 0.1667| ≈ 0.0028
- For 4: |(9/61) - (1/6)| = |0.1475 - 0.1667| ≈ 0.0192
- For 6: |(6/61) - (1/6)| = |0.0984 - 0.1667| ≈ 0.0683
- For 8: |(15/61) - (1/6)| = |0.2459 - 0.1667| ≈ 0.0792
- For 10: |(13/61) - (1/6)| = |0.2131 - 0.1667| ≈ 0.0464
- For 12: |(8/61) - (1/6)| = |0.1311 - 0.1667| ≈ 0.0356
Step 5: Determine the smallest discrepancy.
The smallest discrepancy is:
- For outcome 2: 0.0028
- For outcome 4: 0.0192
- For outcome 6: 0.0683
- For outcome 8: 0.0792
- For outcome 10: 0.0464
- For outcome 12: 0.0356
The smallest discrepancy is 0.0028.
Final Answer: 0.003 (rounded to three decimal places)