Outcomes. 2. 4. 6. 8. 10. 12
Frequency 10 9. 6. 15 13. 8
What is the smallest discrepancy between the experiment and the expected probability of this experiment? Write answer in a 3 decimal places, rounded to the nearest thousandth.
To find the expected probability for each outcome, we divide the total frequency (which is the sum of all frequencies) by 6 (since there are 6 possible outcomes):
Expected probability of getting 2: (10+9+6+15+13+8)/6 = 4.1666667
Expected probability of getting 4: (10+9+6+15+13+8)/6 = 4.1666667
Expected probability of getting 6: (10+9+6+15+13+8)/6 = 4.1666667
Expected probability of getting 8: (10+9+6+15+13+8)/6 = 4.1666667
Expected probability of getting 10: (10+9+6+15+13+8)/6 = 4.1666667
Expected probability of getting 12: (10+9+6+15+13+8)/6 = 4.1666667
To find the discrepancy, we subtract the expected probability from the actual frequency for each outcome, take the absolute value, and add them up:
Discrepancy for outcome 2: |10 - 4.1666667| = 5.8333333
Discrepancy for outcome 4: |9 - 4.1666667| = 4.8333333
Discrepancy for outcome 6: |6 - 4.1666667| = 1.8333333
Discrepancy for outcome 8: |15 - 4.1666667| = 10.8333333
Discrepancy for outcome 10: |13 - 4.1666667| = 8.8333333
Discrepancy for outcome 12: |8 - 4.1666667| = 3.8333333
The smallest discrepancy is 1.8333333, which corresponds to outcome 6. Rounding to the nearest thousandth gives us an answer of 1.833.
To find the smallest discrepancy between the experiment and the expected probability, we need to calculate the expected probability for each outcome and then compare it to the given frequency.
First, calculate the total frequency by summing up the given frequencies:
Total Frequency = 10 + 9 + 6 + 15 + 13 + 8 = 61
Next, calculate the expected probability for each outcome by dividing the frequency of each outcome by the total frequency:
Expected Probability(2) = 10 / 61 ≈ 0.164
Expected Probability(4) = 9 / 61 ≈ 0.148
Expected Probability(6) = 6 / 61 ≈ 0.098
Expected Probability(8) = 15 / 61 ≈ 0.246
Expected Probability(10) = 13 / 61 ≈ 0.213
Expected Probability(12) = 8 / 61 ≈ 0.131
Now, subtract the expected probability from the frequency for each outcome:
Discrepancy(2) = | Frequency(2) - Expected Probability(2) | = | 10 - 0.164 | ≈ 9.836
Discrepancy(4) = | Frequency(4) - Expected Probability(4) | = | 9 - 0.148 | ≈ 8.852
Discrepancy(6) = | Frequency(6) - Expected Probability(6) | = | 6 - 0.098 | ≈ 5.902
Discrepancy(8) = | Frequency(8) - Expected Probability(8) | = | 15 - 0.246 | ≈ 14.754
Discrepancy(10) = | Frequency(10) - Expected Probability(10) | = | 13 - 0.213 | ≈ 12.787
Discrepancy(12) = | Frequency(12) - Expected Probability(12) | = | 8 - 0.131 | ≈ 7.869
The smallest discrepancy is 5.902, which is the discrepancy for the outcome 6.
Therefore, the answer is 5.902.