Outcomes 2 4 6 8 10 12
Frequency 10 9 6 15 13 8
Jede tossed a cube with faces numbered with 2, 4, 6, 8, 10, and 12. The results are recorded in the table. What is the largest discrepancy between the experimental and the expected probability of this experiment? The answer needs to be in percent form to the nearest whole number.
To find the largest discrepancy between the experimental and expected probability, we need to calculate the expected probability and the experimental probability for each outcome.
1. Calculate the total frequency:
Total frequency = 10 + 9 + 6 + 15 + 13 + 8 = 61
2. Calculate the experimental probability for each outcome:
Experimental probability = Frequency / Total frequency
Outcome 2:
Experimental probability = 10 / 61 ≈ 0.164
Outcome 4:
Experimental probability = 9 / 61 ≈ 0.148
Outcome 6:
Experimental probability = 6 / 61 ≈ 0.098
Outcome 8:
Experimental probability = 15 / 61 ≈ 0.246
Outcome 10:
Experimental probability = 13 / 61 ≈ 0.213
Outcome 12:
Experimental probability = 8 / 61 ≈ 0.131
3. Calculate the expected probability for each outcome:
Expected probability = 1 / Total number of outcomes = 1 / 6 ≈ 0.167
4. Calculate the discrepancy for each outcome:
Discrepancy = | Experimental probability - Expected probability |
Outcome 2: | 0.164 - 0.167 | ≈ 0.003
Outcome 4: | 0.148 - 0.167 | ≈ 0.019
Outcome 6: | 0.098 - 0.167 | ≈ 0.069
Outcome 8: | 0.246 - 0.167 | ≈ 0.079
Outcome 10: | 0.213 - 0.167 | ≈ 0.046
Outcome 12: | 0.131 - 0.167 | ≈ 0.036
5. Find the largest discrepancy:
The largest discrepancy is 0.079.
6. Convert the discrepancy to a percentage:
Largest discrepancy in percent form = 0.079 * 100 ≈ 7.9%.
Therefore, the largest discrepancy between the experimental and expected probability in this experiment is 7.9% to the nearest whole number.
To find the experimental probability of each outcome, we divide the frequency by the total number of trials, which is:
10 + 9 + 6 + 15 + 13 + 8 = 61
Experimental Probability of 2 = 10/61 ≈ 0.164
Experimental Probability of 4 = 9/61 ≈ 0.148
Experimental Probability of 6 = 6/61 ≈ 0.098
Experimental Probability of 8 = 15/61 ≈ 0.246
Experimental Probability of 10 = 13/61 ≈ 0.213
Experimental Probability of 12 = 8/61 ≈ 0.131
To find the expected probability of each outcome, we divide 1 by the number of possible outcomes, which is:
1/6 ≈ 0.167
Expected Probability of 2 = 0.167
Expected Probability of 4 = 0.167
Expected Probability of 6 = 0.167
Expected Probability of 8 = 0.167
Expected Probability of 10 = 0.167
Expected Probability of 12 = 0.167
To find the discrepancy between the experimental and expected probabilities, we subtract the expected probability from the experimental probability for each outcome, and take the absolute value of the difference. Then we find the largest discrepancy:
|Experimental Probability of 2 - Expected Probability of 2| = |0.164 - 0.167| ≈ 0.003
|Experimental Probability of 4 - Expected Probability of 4| = |0.148 - 0.167| ≈ 0.019
|Experimental Probability of 6 - Expected Probability of 6| = |0.098 - 0.167| ≈ 0.069
|Experimental Probability of 8 - Expected Probability of 8| = |0.246 - 0.167| ≈ 0.079
|Experimental Probability of 10 - Expected Probability of 10| = |0.213 - 0.167| ≈ 0.046
|Experimental Probability of 12 - Expected Probability of 12| = |0.131 - 0.167| ≈ 0.036
The largest discrepancy is 0.079, which corresponds to the outcome of 8. To find the discrepancy as a percentage, we divide it by the expected probability and multiply by 100:
0.079/0.167 x 100 ≈ 47 (rounded to the nearest whole number)
Therefore, the largest discrepancy between the experimental and expected probability of this experiment is 47%.
To find the largest discrepancy between the experimental and expected probability, we need to calculate the expected probability for each outcome and compare it with the experimental probability.
1. First, let's calculate the total number of trials (n) by summing up the frequencies:
n = 10 + 9 + 6 + 15 + 13 + 8 = 61
2. Next, let's calculate the expected probability for each outcome. Since the cube has 6 faces numbered 2, 4, 6, 8, 10, and 12, the probability of getting each number is 1/6 or approximately 0.1667.
Expected probability = 0.1667
3. Now, we can calculate the expected frequency for each outcome by multiplying the expected probability by the total number of trials (n):
Expected frequency = Expected probability × n
For each outcome, the expected frequency is:
2: 0.1667 × 61 = 10.17 (rounded to nearest whole number: 10)
4: 0.1667 × 61 = 10.17 (rounded to nearest whole number: 10)
6: 0.1667 × 61 = 10.17 (rounded to nearest whole number: 10)
8: 0.1667 × 61 = 10.17 (rounded to nearest whole number: 10)
10: 0.1667 × 61 = 10.17 (rounded to nearest whole number: 10)
12: 0.1667 × 61 = 10.17 (rounded to nearest whole number: 10)
4. Then, we compare the expected frequency with the experimental frequency for each outcome to find the discrepancy. The discrepancy is the absolute difference between the experimental frequency and the expected frequency.
For each outcome:
Discrepancy = |Experimental frequency - Expected frequency|
For example, for the first outcome:
Outcome: 2
Experimental frequency = 10
Expected frequency = 10
Discrepancy = |10 - 10| = 0
Calculating the discrepancies for all outcomes, we get:
Outcome: 2, Discrepancy = 0
Outcome: 4, Discrepancy = |9 - 10| = 1
Outcome: 6, Discrepancy = |6 - 10| = 4
Outcome: 8, Discrepancy = |15 - 10| = 5
Outcome: 10, Discrepancy = |13 - 10| = 3
Outcome: 12, Discrepancy = |8 - 10| = 2
5. Finally, we find the largest discrepancy by comparing all the discrepancies calculated above. The largest discrepancy is 5 (corresponding to outcome 8).
Therefore, the largest discrepancy between the experimental and the expected probability is 5, expressed as a percent to the nearest whole number: 5%.