To find the smallest discrepancy between the experimental and expected probabilities, we first need to calculate the expected frequency for each outcome, and then compare it with the experimental frequencies provided.
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Calculate the total number of trials (N): \[ N = 10 + 9 + 6 + 15 + 13 + 8 = 61 \]
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Determine the expected probability for each outcome. Since the cube has 6 faces, the expected probability for each number (assuming a fair cube) should be: \[ P(\text{each outcome}) = \frac{1}{6} \] Therefore, the expected frequency for each outcome is: \[ \text{Expected frequency} = N \times P(\text{each outcome}) = 61 \times \frac{1}{6} \approx 10.167 \]
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Calculate the experimental probabilities for each outcome:
- For 2: \( \frac{10}{61} \approx 0.164 \)
- For 4: \( \frac{9}{61} \approx 0.148 \)
- For 6: \( \frac{6}{61} \approx 0.098 \)
- For 8: \( \frac{15}{61} \approx 0.246 \)
- For 10: \( \frac{13}{61} \approx 0.213 \)
- For 12: \( \frac{8}{61} \approx 0.131 \)
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Calculate the expected probability for each outcome, which is \( \frac{1}{6} \approx 0.167 \).
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Calculate discrepancies for each outcome by finding the absolute difference between experimental probabilities and expected probabilities:
- For 2: \( |0.164 - 0.167| \approx 0.003 \)
- For 4: \( |0.148 - 0.167| \approx 0.019 \)
- For 6: \( |0.098 - 0.167| \approx 0.069 \)
- For 8: \( |0.246 - 0.167| \approx 0.079 \)
- For 10: \( |0.213 - 0.167| \approx 0.046 \)
- For 12: \( |0.131 - 0.167| \approx 0.036 \)
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Find the smallest discrepancy:
- The smallest discrepancy is \( 0.003 \) from the outcome \( 2 \).
Thus, the smallest discrepancy between the experimental and expected probabilities in this experiment is:
\[ \boxed{0.003} \]