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 find the expected probability for each outcome, we need to consider that each face has an equal probability of \(\frac{1}{6}\) since there are 6 faces on the cube.
Therefore, the expected probability for outcomes 2, 4, 6, 8, 10, and 12 are all \(\frac{1}{6}\).
Now, we can calculate the discrepancy for each outcome by finding the absolute difference between the experimental frequency and the expected probability.
| Outcome | Frequency | Expected Probability | Discrepancy |
|---------|-----------|-----------------------|-------------|
| 2 | 10 | \(\frac{1}{6}\) | \(|10 - \frac{1}{6}| = \frac{59}{6}\) |
| 4 | 9 | \(\frac{1}{6}\) | \(|9 - \frac{1}{6}| = \frac{53}{6}\) |
| 6 | 6 | \(\frac{1}{6}\) | \(|6 - \frac{1}{6}| = \frac{35}{6}\) |
| 8 | 15 | \(\frac{1}{6}\) | \(|15 - \frac{1}{6}| = \frac{89}{6}\) |
| 10 | 13 | \(\frac{1}{6}\) | \(|13 - \frac{1}{6}| = \frac{77}{6}\) |
| 12 | 8 | \(\frac{1}{6}\) | \(|8 - \frac{1}{6}| = \frac{47}{6}\) |
Now, we find the smallest discrepancy which is \(\frac{35}{6}\).
Converting this to a decimal rounded to the nearest thousandth, the smallest discrepancy between the experimental and expected probability is approximately 5.833.