To find the largest discrepancy between the experimental and expected probabilities for the outcomes of the dice toss experiment, we'll first calculate both the experimental probabilities based on the given frequencies and the expected probabilities based on the theoretical outcomes.
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Calculate the total number of tosses:
Total = \(10 + 9 + 6 + 15 + 13 + 8 = 61\) -
Calculate the experimental probabilities:
The experimental probability for each outcome is calculated as: \[ P_{\text{experimental}}(x) = \frac{\text{Frequency of } x}{\text{Total Frequencies}} \]- For outcome 2: \[ P_{\text{experimental}}(2) = \frac{10}{61} \approx 0.164 \]
- For outcome 4: \[ P_{\text{experimental}}(4) = \frac{9}{61} \approx 0.148 \]
- For outcome 6: \[ P_{\text{experimental}}(6) = \frac{6}{61} \approx 0.098 \]
- For outcome 8: \[ P_{\text{experimental}}(8) = \frac{15}{61} \approx 0.246 \]
- For outcome 10: \[ P_{\text{experimental}}(10) = \frac{13}{61} \approx 0.213 \]
- For outcome 12: \[ P_{\text{experimental}}(12) = \frac{8}{61} \approx 0.131 \]
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Calculate the expected probabilities:
For a fair die with outcomes 2, 4, 6, 8, 10, and 12, each outcome has an equal probability of occurrence: \[ P_{\text{expected}}(x) = \frac{1}{6} \approx 0.167 \] -
Calculate the discrepancies:
The discrepancy is calculated as: \[ \text{Discrepancy} = |P_{\text{experimental}}(x) - P_{\text{expected}}(x)| \] Now we calculate the discrepancies for each outcome:
- For 2: \[ \text{Discrepancy} = |0.164 - 0.167| = 0.003 \]
- For 4: \[ \text{Discrepancy} = |0.148 - 0.167| = 0.019 \]
- For 6: \[ \text{Discrepancy} = |0.098 - 0.167| = 0.069 \]
- For 8: \[ \text{Discrepancy} = |0.246 - 0.167| = 0.079 \]
- For 10: \[ \text{Discrepancy} = |0.213 - 0.167| = 0.046 \]
- For 12: \[ \text{Discrepancy} = |0.131 - 0.167| = 0.036 \]
- Find the largest discrepancy:
The largest discrepancy is: \[ 0.079 \]
Finally, rounding to the nearest thousandth, the largest discrepancy is: \[ \boxed{0.079} \]