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 expected probability, we need to divide each frequency by the total number of trials, which is:
10 + 9 + 6 + 15 + 13 + 8 = 61
Expected probability of getting 2 = 10/61 = 0.1639
Expected probability of getting 4 = 9/61 = 0.1475
Expected probability of getting 6 = 6/61 = 0.0984
Expected probability of getting 8 = 15/61 = 0.2459
Expected probability of getting 10 = 13/61 = 0.2131
Expected probability of getting 12 = 8/61 = 0.1311
To calculate the discrepancy between the experimental and expected probability, we need to subtract the expected probability from the experimental probability and take the absolute value. Then, we take the maximum value of these discrepancies.
Discrepancy for 2 = |0.1639 - 10/61| = 0.0003
Discrepancy for 4 = |0.1475 - 9/61| = 0.0008
Discrepancy for 6 = |0.0984 - 6/61| = 0.0002
Discrepancy for 8 = |0.2459 - 15/61| = 0.0002
Discrepancy for 10 = |0.2131 - 13/61| = 0.0001
Discrepancy for 12 = |0.1311 - 8/61| = 0.0003
The smallest discrepancy is 0.0001, which occurs for the outcome of 10. Therefore, the answer is 0.000, rounded to the nearest thousandth.
To find the smallest discrepancy between the experimental and expected probability, we need to compare the observed frequencies with the expected probabilities for each outcome.
The expected probability for each outcome can be calculated by dividing the frequency by the total number of observations. The total number of observations can be found by summing up all the frequencies.
Let's calculate the expected probabilities for each outcome first:
Outcome 2:
Expected probability = Frequency / Total observations
= 10 / (10 + 9 + 6 + 15 + 13 + 8)
= 10 / 61
≈ 0.164
Outcome 4:
Expected probability = Frequency / Total observations
= 9 / 61
≈ 0.148
Outcome 6:
Expected probability = Frequency / Total observations
= 6 / 61
≈ 0.098
Outcome 8:
Expected probability = Frequency / Total observations
= 15 / 61
≈ 0.246
Outcome 10:
Expected probability = Frequency / Total observations
= 13 / 61
≈ 0.213
Outcome 12:
Expected probability = Frequency / Total observations
= 8 / 61
≈ 0.131
Now we can calculate the discrepancies between the experimental and expected probabilities for each outcome:
Outcome 2:
Discrepancy = |Expected probability - Observed probability|
= |0.164 - (10 / 61)|
≈ 0.000
Outcome 4:
Discrepancy = |0.148 - (9 / 61)|
≈ 0.000
Outcome 6:
Discrepancy = |0.098 - (6 / 61)|
≈ 0.000
Outcome 8:
Discrepancy = |0.246 - (15 / 61)|
≈ 0.000
Outcome 10:
Discrepancy = |0.213 - (13 / 61)|
≈ 0.000
Outcome 12:
Discrepancy = |0.131 - (8 / 61)|
≈ 0.000
From the calculations, we can see that the smallest discrepancy between the experimental and expected probability is approximately 0.000.