To find the outcome value with the largest discrepancy between the experimental and expected probability, we first need to calculate the experimental probabilities from the frequency data.
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Calculating total number of outcomes: We add up the frequencies: \[ \text{Total Outcomes} = 10 + 9 + 6 + 15 + 13 + 8 = 61 \]
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Calculating experimental probabilities: We find the probability for each outcome by dividing its frequency by the total number of outcomes: \[ P(\text{outcome}) = \frac{\text{Frequency}}{\text{Total Outcomes}} \]
- For 2: \[ P(2) = \frac{10}{61} \approx 0.1639 \]
- For 4: \[ P(4) = \frac{9}{61} \approx 0.1475 \]
- For 6: \[ P(6) = \frac{6}{61} \approx 0.0984 \]
- For 8: \[ P(8) = \frac{15}{61} \approx 0.2459 \]
- For 10: \[ P(10) = \frac{13}{61} \approx 0.2131 \]
- For 12: \[ P(12) = \frac{8}{61} \approx 0.1311 \]
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Calculating expected probabilities: Since the cube has 6 faces with numbers 2, 4, 6, 8, 10, and 12, we assume each outcome is equally likely. Hence, the expected probability for each outcome is: \[ P(\text{expected}) = \frac{1}{6} \approx 0.1667 \]
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Calculating discrepancies: We now calculate the discrepancy (absolute difference) between experimental and expected probabilities:
- For 2: \[ |0.1639 - 0.1667| \approx 0.0028 \]
- For 4: \[ |0.1475 - 0.1667| \approx 0.0192 \]
- For 6: \[ |0.0984 - 0.1667| \approx 0.0683 \]
- For 8: \[ |0.2459 - 0.1667| \approx 0.0792 \]
- For 10: \[ |0.2131 - 0.1667| \approx 0.0464 \]
- For 12: \[ |0.1311 - 0.1667| \approx 0.0356 \]
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Identifying the largest discrepancy: The largest discrepancy is for the outcome of 8, which has a discrepancy of approximately 0.0792.
Thus, the outcome value with the largest discrepancy between the experimental and expected probability is 8.