To find the smallest discrepancy between the experimental and expected probabilities, we first need to calculate both the experimental probability for each outcome and the expected probability for each outcome.
Step 1: Calculate Total Frequency
Let's first determine the total number of occurrences (total tosses).
\[ \text{Total Frequency} = 10 + 9 + 6 + 15 + 13 + 8 = 61 \]
Step 2: Calculate Experimental Probability
The experimental probability for each outcome is calculated as:
\[ P(E) = \frac{\text{Frequency}}{\text{Total Frequency}} \]
Calculating for each outcome:
- For 2: \[ P(2) = \frac{10}{61} \approx 0.164 \]
- For 4: \[ P(4) = \frac{9}{61} \approx 0.148 \]
- For 6: \[ P(6) = \frac{6}{61} \approx 0.098 \]
- For 8: \[ P(8) = \frac{15}{61} \approx 0.246 \]
- For 10: \[ P(10) = \frac{13}{61} \approx 0.213 \]
- For 12: \[ P(12) = \frac{8}{61} \approx 0.131 \]
Step 3: Calculate Expected Probability
Since the cube is fair and has 6 outcomes, the expected probability for each outcome is:
\[ P(E) = \frac{1}{6} \approx 0.167 \]
Step 4: Calculate Discrepancies
Next, we calculate the discrepancies between the experimental probabilities and the expected probability:
- For 2: \[ |P(2) - P(E)| = \left| 0.164 - 0.167 \right| \approx 0.003 \]
- For 4: \[ |P(4) - P(E)| = \left| 0.148 - 0.167 \right| \approx 0.019 \]
- For 6: \[ |P(6) - P(E)| = \left| 0.098 - 0.167 \right| \approx 0.069 \]
- For 8: \[ |P(8) - P(E)| = \left| 0.246 - 0.167 \right| \approx 0.079 \]
- For 10: \[ |P(10) - P(E)| = \left| 0.213 - 0.167 \right| \approx 0.046 \]
- For 12: \[ |P(12) - P(E)| = \left| 0.131 - 0.167 \right| \approx 0.036 \]
Step 5: Find the Smallest Discrepancy
Now we compare all discrepancies calculated:
- For 2: 0.003
- For 4: 0.019
- For 6: 0.069
- For 8: 0.079
- For 10: 0.046
- For 12: 0.036
The smallest discrepancy is:
\[ \text{Smallest Discrepancy} = 0.003 \]
Step 6: Final Result
Rounding to three decimal places, we write:
\[ \boxed{0.003} \]