Outcomes 2. 4. 6. 8. 10. 12
Frequency 10 9. 6. 15. 13. 8
What is the largest discrepancy between the experiment and the expected probability of this experiment?
To determine the expected probability, we need to calculate the probability of each outcome based on a fair die:
Outcome: 2. Probability: 1/6
Outcome: 4. Probability: 1/6
Outcome: 6. Probability: 1/6
Outcome: 8. Probability: 1/6
Outcome: 10. Probability: 1/6
Outcome: 12. Probability: 1/6
The expected frequency of each outcome can be calculated by multiplying the probability of each outcome by the total number of trials (in this case, the sum of the frequencies):
Expected frequency of outcome 2: (1/6) x 61 ≈ 10.17
Expected frequency of outcome 4: (1/6) x 61 ≈ 10.17
Expected frequency of outcome 6: (1/6) x 61 ≈ 10.17
Expected frequency of outcome 8: (1/6) x 61 ≈ 10.17
Expected frequency of outcome 10: (1/6) x 61 ≈ 10.17
Expected frequency of outcome 12: (1/6) x 61 ≈ 10.17
To find the largest discrepancy between the experiment and the expected probability, we need to compare the actual frequency of each outcome to its expected frequency. The largest discrepancy will occur for the outcome with the largest absolute difference between its actual frequency and expected frequency.
Outcome 2: Actual frequency = 10, expected frequency ≈ 10.17, absolute difference = 0.17
Outcome 4: Actual frequency = 9, expected frequency ≈ 10.17, absolute difference = 1.17
Outcome 6: Actual frequency = 6, expected frequency ≈ 10.17, absolute difference = 4.17
Outcome 8: Actual frequency = 15, expected frequency ≈ 10.17, absolute difference = 4.83
Outcome 10: Actual frequency = 13, expected frequency ≈ 10.17, absolute difference = 2.83
Outcome 12: Actual frequency = 8, expected frequency ≈ 10.17, absolute difference = 2.17
The largest discrepancy is 4.83, which occurs for outcome 8. Therefore, the largest discrepancy between the experiment and the expected probability of this experiment is 4.83.
To calculate the largest discrepancy between the experiment and the expected probability, we need to find the expected probability for each outcome and then compare it with the actual frequency observed in the experiment.
Step 1: Calculate the total frequency
Total frequency = Sum of all frequencies = 10 + 9 + 6 + 15 + 13 + 8 = 61
Step 2: Calculate the expected probability for each outcome
Expected probability = Frequency / Total frequency
For outcome 2:
Expected probability = 10 / 61 ≈ 0.1639
For outcome 4:
Expected probability = 9 / 61 ≈ 0.1475
For outcome 6:
Expected probability = 6 / 61 ≈ 0.0984
For outcome 8:
Expected probability = 15 / 61 ≈ 0.2459
For outcome 10:
Expected probability = 13 / 61 ≈ 0.2131
For outcome 12:
Expected probability = 8 / 61 ≈ 0.1311
Step 3: Calculate the discrepancy for each outcome
Discrepancy = |Actual frequency - Expected probability|
For outcome 2:
Discrepancy = |10 - 0.1639| ≈ 9.8361
For outcome 4:
Discrepancy = |9 - 0.1475| ≈ 8.8525
For outcome 6:
Discrepancy = |6 - 0.0984| ≈ 5.9016
For outcome 8:
Discrepancy = |15 - 0.2459| ≈ 14.7541
For outcome 10:
Discrepancy = |13 - 0.2131| ≈ 12.7869
For outcome 12:
Discrepancy = |8 - 0.1311| ≈ 7.8689
Step 4: Find the largest discrepancy
The largest discrepancy is the highest value among the calculated discrepancies. In this case, the largest discrepancy is 14.7541, which occurs for outcome 8.
Therefore, the largest discrepancy between the experiment and the expected probability of this experiment is approximately 14.7541.