To find the largest discrepancy between the experimental and expected probabilities for the outcomes of Jede's experiment, we first need to calculate both the experimental and expected probabilities for each outcome.
-
Calculate the total number of trials (N): \[ N = 10 + 9 + 6 + 15 + 13 + 8 = 61 \]
-
Calculate the experimental probability (P_e) for each outcome: \[ P_e(\text{outcome}) = \frac{\text{Frequency}}{N} \]
| Outcomes | Frequency | Experimental Probability | |----------|-----------|-------------------------| | 2 | 10 | \( P_e(2) = \frac{10}{61} \approx 0.16393 \) | | 4 | 9 | \( P_e(4) = \frac{9}{61} \approx 0.14754 \) | | 6 | 6 | \( P_e(6) = \frac{6}{61} \approx 0.09836 \) | | 8 | 15 | \( P_e(8) = \frac{15}{61} \approx 0.24590 \) | | 10 | 13 | \( P_e(10) = \frac{13}{61} \approx 0.21311 \) | | 12 | 8 | \( P_e(12) = \frac{8}{61} \approx 0.13115 \) |
-
Calculate the expected probability (P_e) for each outcome: Since the cube is fair and has 6 faces, the expected probability for each outcome is: \[ P_e(\text{outcome}) = \frac{1}{6} \approx 0.16667 \]
-
Calculate the discrepancies: Now we compute the difference between the experimental probability and the expected probability, which gives us the discrepancy for each outcome:
\[ \text{Discrepancy} = \left| P_e(\text{experimental}) - P_e(\text{expected}) \right| \]
| Outcomes | Experimental Probability | Expected Probability | Discrepancy | |----------|-------------------------|---------------------|---------------------------| | 2 | 0.16393 | 0.16667 | \( |0.16393 - 0.16667| \approx 0.00274 \) | | 4 | 0.14754 | 0.16667 | \( |0.14754 - 0.16667| \approx 0.01913 \) | | 6 | 0.09836 | 0.16667 | \( |0.09836 - 0.16667| \approx 0.06831 \) | | 8 | 0.24590 | 0.16667 | \( |0.24590 - 0.16667| \approx 0.07923 \) | | 10 | 0.21311 | 0.16667 | \( |0.21311 - 0.16667| \approx 0.04644 \) | | 12 | 0.13115 | 0.16667 | \( |0.13115 - 0.16667| \approx 0.03552 \) |
-
Find the largest discrepancy: The largest discrepancy occurs at the outcome of 8, which is: \[ 0.07923 \]
-
Convert the largest discrepancy to percentage: \[ \text{Largest Discrepancy (percent)} = 0.07923 \times 100 \approx 7.923% \]
Rounding to the nearest whole number, we get:
\[ \text{Largest Discrepancy} \approx 8% \]
Thus, the largest discrepancy between the experimental and expected probabilities is \(\boxed{8}%\).