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.
To calculate the experimental probability, we need to divide the frequency of each outcome by the total number of trials.
The total number of trials is the sum of the frequencies:
10 + 9 + 6 + 15 + 13 + 8 = 61
Now, let's calculate the experimental probability for each outcome:
Outcome 2: 10/61 ≈ 0.164
Outcome 4: 9/61 ≈ 0.148
Outcome 6: 6/61 ≈ 0.098
Outcome 8: 15/61 ≈ 0.246
Outcome 10: 13/61 ≈ 0.213
Outcome 12: 8/61 ≈ 0.131
The expected probability for each outcome is 1/6 since there are 6 possible outcomes and each face of the dice has an equal chance of being rolled.
Now, let's calculate the discrepancy for each outcome by subtracting the expected probability from the experimental probability:
Outcome 2: 0.164 - 1/6 ≈ -0.014
Outcome 4: 0.148 - 1/6 ≈ -0.018
Outcome 6: 0.098 - 1/6 ≈ -0.068
Outcome 8: 0.246 - 1/6 ≈ 0.080
Outcome 10: 0.213 - 1/6 ≈ 0.047
Outcome 12: 0.131 - 1/6 ≈ -0.018
The smallest discrepancy is -0.068, which corresponds to the outcome 6.
Therefore, the smallest discrepancy between the experimental and expected probability of this experiment is -0.068.
To find the smallest discrepancy between the experimental and expected probability of this experiment, we need to calculate the expected probability for each outcome and then compare it with the experimental frequency.
The expected probability for each outcome is calculated by dividing the frequency of each outcome by the total number of trials. In this case, the total number of trials is the sum of all frequencies, which is 10+9+6+15+13+8=61.
Expected probability for outcome 2:
Probability = Frequency of outcome 2 / Total number of trials
Probability = 10 / 61
Probability ≈ 0.164
Expected probability for outcome 4:
Probability = Frequency of outcome 4 / Total number of trials
Probability = 9 / 61
Probability ≈ 0.148
Similarly, we can calculate the expected probabilities for the other outcomes:
Outcome 6: Probability ≈ 0.098
Outcome 8: Probability ≈ 0.246
Outcome 10: Probability ≈ 0.213
Outcome 12: Probability ≈ 0.131
Now, we can calculate the discrepancy for each outcome by subtracting the experimental frequency from the expected probability. The discrepancy measures how much the experimental result deviates from the expected outcome.
Discrepancy for outcome 2: 0.164 - 10 = -9.836
Discrepancy for outcome 4: 0.148 - 9 = -8.852
Discrepancy for outcome 6: 0.098 - 6 = -5.902
Discrepancy for outcome 8: 0.246 - 15 = -14.754
Discrepancy for outcome 10: 0.213 - 13 = -12.787
Discrepancy for outcome 12: 0.131 - 8 = -7.869
To find the smallest discrepancy, we need to find the absolute value of each discrepancy and select the smallest one.
The absolute values of the discrepancies are:
9.836, 8.852, 5.902, 14.754, 12.787, 7.869
The smallest discrepancy is 5.902, which corresponds to the outcome 6.
Therefore, the smallest discrepancy between the experimental and expected probability of this experiment is 5.902 (rounded to three decimal places).
First, we need to find the expected probability for each outcome. Since there are 6 equally likely outcomes on the cube, each outcome has a probability of 1/6. Therefore, the expected frequencies are:
Outcome: 2 4 6 8 10 12
Expected frequency: 15 15 15 15 15 15
To find the discrepancy between the experimental and expected probabilities, we can use the formula:
|Observed frequency - Expected frequency| / Expected frequency
We can calculate this for each outcome and take the maximum value as the discrepancy:
Outcome: 2 4 6 8 10 12
Frequency: 10 9 6 15 13 8
Expected frequency: 15 15 15 15 15 15
Discrepancy: 0.333 0.400 0.600 0 0.133 0.467
The smallest discrepancy is 0, which occurs for outcome 8. Therefore, the answer is 0 (rounded to the nearest thousandth).
Well, let's start by calculating the experimental probability for each outcome. The total number of trials is the sum of the frequencies, which is 10+9+6+15+13+8 = 61.
Experimental probability for a 2: 10/61 ≈ 0.164
Experimental probability for a 4: 9/61 ≈ 0.148
Experimental probability for a 6: 6/61 ≈ 0.098
Experimental probability for an 8: 15/61 ≈ 0.246
Experimental probability for a 10: 13/61 ≈ 0.213
Experimental probability for a 12: 8/61 ≈ 0.131
Now let's calculate the expected probability for each outcome. Since we have a fair cube, each face has an equal chance of being rolled. There are 6 faces, so the expected probability for each outcome is 1/6 ≈ 0.167.
Now let's calculate the discrepancy for each outcome by subtracting the expected probability from the experimental probability:
Discrepancy for a 2: 0.164 - 0.167 ≈ -0.003
Discrepancy for a 4: 0.148 - 0.167 ≈ -0.019
Discrepancy for a 6: 0.098 - 0.167 ≈ -0.069
Discrepancy for an 8: 0.246 - 0.167 ≈ 0.079
Discrepancy for a 10: 0.213 - 0.167 ≈ 0.046
Discrepancy for a 12: 0.131 - 0.167 ≈ -0.036
The smallest discrepancy is -0.069, which corresponds to the outcome 6. So, the smallest discrepancy between the experimental and expected probability is approximately -0.069, rounded to the nearest thousandth.
Hope that helps, or at least adds a touch of humor to your math question!