Use the table to answer the question.
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 smallest discrepancy between the experimental and the expected probability of this experiment? Write your answer in 3 decimal places, rounded to the nearest thousandth.
(1 point)
To find the smallest discrepancy between the experimental and expected probability, we first need to calculate the expected probability for each outcome.
The total number of outcomes is the sum of the frequencies, which is:
Total outcomes = 10 + 9 + 6 + 15 + 13 + 8 = 61
The probability for each outcome is the frequency divided by the total number of outcomes. We can calculate this for each outcome:
Probability for 2: 10/61 ≈ 0.164
Probability for 4: 9/61 ≈ 0.148
Probability for 6: 6/61 ≈ 0.098
Probability for 8: 15/61 ≈ 0.246
Probability for 10: 13/61 ≈ 0.213
Probability for 12: 8/61 ≈ 0.131
The expected probability for each outcome is:
Expected probability for 2: 1/6 ≈ 0.167
Expected probability for 4: 1/6 ≈ 0.167
Expected probability for 6: 1/6 ≈ 0.167
Expected probability for 8: 1/6 ≈ 0.167
Expected probability for 10: 1/6 ≈ 0.167
Expected probability for 12: 1/6 ≈ 0.167
To find the discrepancy between the experimental and expected probability for each outcome, we subtract the expected probability from the experimental probability:
Discrepancy for 2: 0.164 - 0.167 = -0.003
Discrepancy for 4: 0.148 - 0.167 = -0.019
Discrepancy for 6: 0.098 - 0.167 = -0.069
Discrepancy for 8: 0.246 - 0.167 = 0.079
Discrepancy for 10: 0.213 - 0.167 = 0.046
Discrepancy for 12: 0.131 - 0.167 = -0.036
The smallest discrepancy is the smallest absolute value among these values, which is 0.003 (rounded to three decimal places).
Therefore, the smallest discrepancy between the experimental and expected probability is 0.003.
Everyone! don't listen to Explain bot or Bot! Step by Step bot is correct the answer is 0.003 on the practice!
The expected probability for each outcome is 1/6. We can calculate the experimental probability by dividing the frequency of each outcome by the total number of trials:
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
The discrepancies between the experimental and expected probabilities are:
|0.164 - 1/6| ≈ 0.014
|0.148 - 1/6| ≈ 0.012
|0.098 - 1/6| ≈ 0.068
|0.246 - 1/6| ≈ 0.080
|0.213 - 1/6| ≈ 0.047
|0.131 - 1/6| ≈ 0.034
The smallest discrepancy is 0.012, so the answer is 0.012 rounded to the nearest thousandth.
Answer: 0.012
To find the smallest discrepancy between the experimental and expected probability, we need to calculate the expected probability and then compare it to the experimental probability for each outcome.
1. Start by calculating the total frequency (sum of all the frequencies in the table). In this case, the total frequency is:
10 + 9 + 6 + 15 + 13 + 8 = 61
2. Next, calculate the expected probability for each outcome. The expected probability is calculated by dividing the frequency of each outcome by the total frequency.
For example:
- For outcome 2: Expected probability = Frequency of 2 / Total frequency = 10 / 61 ≈ 0.164
- For outcome 4: Expected probability = Frequency of 4 / Total frequency = 9 / 61 ≈ 0.148
- For outcome 6: Expected probability = Frequency of 6 / Total frequency = 6 / 61 ≈ 0.098
- For outcome 8: Expected probability = Frequency of 8 / Total frequency = 15 / 61 ≈ 0.246
- For outcome 10: Expected probability = Frequency of 10 / Total frequency = 13 / 61 ≈ 0.213
- For outcome 12: Expected probability = Frequency of 12 / Total frequency = 8 / 61 ≈ 0.131
3. Now, we can calculate the experimental probability for each outcome by dividing the frequency of each outcome by the total frequency.
For example:
- For outcome 2: Experimental probability = Frequency of 2 / Total frequency = 10 / 61 ≈ 0.164
- For outcome 4: Experimental probability = Frequency of 4 / Total frequency = 9 / 61 ≈ 0.148
- For outcome 6: Experimental probability = Frequency of 6 / Total frequency = 6 / 61 ≈ 0.098
- For outcome 8: Experimental probability = Frequency of 8 / Total frequency = 15 / 61 ≈ 0.246
- For outcome 10: Experimental probability = Frequency of 10 / Total frequency = 13 / 61 ≈ 0.213
- For outcome 12: Experimental probability = Frequency of 12 / Total frequency = 8 / 61 ≈ 0.131
4. Finally, we calculate the discrepancy between the experimental and expected probabilities for each outcome by subtracting the expected probability from the experimental probability (or vice versa).
For example:
- For outcome 2: Discrepancy = Experimental probability - Expected probability = 0.164 - 0.164 = 0
- For outcome 4: Discrepancy = Experimental probability - Expected probability = 0.148 - 0.148 = 0
- For outcome 6: Discrepancy = Experimental probability - Expected probability = 0.098 - 0.098 = 0
- For outcome 8: Discrepancy = Experimental probability - Expected probability = 0.246 - 0.246 = 0
- For outcome 10: Discrepancy = Experimental probability - Expected probability = 0.213 - 0.213 = 0
- For outcome 12: Discrepancy = Experimental probability - Expected probability = 0.131 - 0.131 = 0
Since all the discrepancies are 0, the smallest discrepancy between the experimental and expected probability is also 0.