To solve the problem, we first need to establish the expected probability for each outcome when rolling the cube. Since the cube has faces numbered with 2, 4, 6, 8, 10, and 12, there are a total of 6 outcomes. Assuming a fair die, the expected probability for each number should be equal.
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Calculate the total number of tosses:
- Total frequency = \( 10 + 9 + 6 + 15 + 13 + 8 = 61 \)
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Calculate the expected probability for each outcome:
- For each of the 6 outcomes (2, 4, 6, 8, 10, 12), the expected probability is: \[ P(\text{each outcome}) = \frac{1}{6} \approx 0.166667 \]
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Calculate the experimental probability for each outcome:
- For outcome 2: \( P(2) = \frac{10}{61} \approx 0.163934 \)
- For outcome 4: \( P(4) = \frac{9}{61} \approx 0.147541 \)
- For outcome 6: \( P(6) = \frac{6}{61} \approx 0.098361 \)
- For outcome 8: \( P(8) = \frac{15}{61} \approx 0.245902 \)
- For outcome 10: \( P(10) = \frac{13}{61} \approx 0.213114 \)
- For outcome 12: \( P(12) = \frac{8}{61} \approx 0.131148 \)
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Calculate the discrepancies for each outcome:
- Discrepancy for outcome 2: \( |0.163934 - 0.166667| \approx 0.002733 \)
- Discrepancy for outcome 4: \( |0.147541 - 0.166667| \approx 0.019126 \)
- Discrepancy for outcome 6: \( |0.098361 - 0.166667| \approx 0.068306 \)
- Discrepancy for outcome 8: \( |0.245902 - 0.166667| \approx 0.079235 \)
- Discrepancy for outcome 10: \( |0.213114 - 0.166667| \approx 0.046447 \)
- Discrepancy for outcome 12: \( |0.131148 - 0.166667| \approx 0.035519 \)
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Find the smallest discrepancy:
- The smallest discrepancy among all results is: \( 0.002733 \) (for outcome 2).
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Round to three decimal places:
- The smallest discrepancy rounded to three decimal places is: \[ \boxed{0.003} \]