To determine which outcome has the largest discrepancy between the experimental and expected probabilities, we first need to calculate both probabilities for each outcome.
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Calculate the total number of outcomes (tosses): \[ \text{Total Frequency} = 10 + 9 + 6 + 15 + 13 + 8 = 61 \]
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Calculate the experimental probability for each outcome: The experimental probability is calculated as the frequency of each outcome divided by the total number of 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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Determine the expected probability for each outcome: Since each outcome on the cube is 2, 4, 6, 8, 10, and 12, and assuming the cube was fair, the expected probability for each would be: \[ P(\text{each outcome}) = \frac{1}{6} \approx 0.1667 \]
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Calculate the discrepancies: Now we will calculate the absolute discrepancy for each outcome, which is the absolute value of the difference between the experimental probability and the expected probability:
- For 2: \[ \left| P(2) - P(\text{expected}) \right| = \left| 0.1639 - 0.1667 \right| \approx 0.0028 \]
- For 4: \[ \left| P(4) - P(\text{expected}) \right| = \left| 0.1475 - 0.1667 \right| \approx 0.0192 \]
- For 6: \[ \left| P(6) - P(\text{expected}) \right| = \left| 0.0984 - 0.1667 \right| \approx 0.0683 \]
- For 8: \[ \left| P(8) - P(\text{expected}) \right| = \left| 0.2459 - 0.1667 \right| \approx 0.0792 \]
- For 10: \[ \left| P(10) - P(\text{expected}) \right| = \left| 0.2131 - 0.1667 \right| \approx 0.0464 \]
- For 12: \[ \left| P(12) - P(\text{expected}) \right| = \left| 0.1311 - 0.1667 \right| \approx 0.0356 \]
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Identify the largest discrepancy: The discrepancies are:
- For 2: 0.0028
- For 4: 0.0192
- For 6: 0.0683
- For 8: 0.0792
- For 10: 0.0464
- For 12: 0.0356
The largest discrepancy is for the outcome 8, with a discrepancy of approximately 0.0792.