Question

Use the table to answer the question.

The table presents the frequency of occurrences for numbers 2, 4, 6, 8, and 10 when Jade tossed a cube.
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)

Answer 1

To find the smallest discrepancy between the experimental and expected probabilities, we first need to calculate the expected frequency for each outcome, and then compare it with the experimental frequencies provided.

1. **Calculate the total number of trials (N)**:
\[
N = 10 + 9 + 6 + 15 + 13 + 8 = 61
\]

2. **Determine the expected probability for each outcome**. Since the cube has 6 faces, the expected probability for each number (assuming a fair cube) should be:
\[
P(\text{each outcome}) = \frac{1}{6}
\]
Therefore, the expected frequency for each outcome is:
\[
\text{Expected frequency} = N \times P(\text{each outcome}) = 61 \times \frac{1}{6} \approx 10.167
\]

3. **Calculate the experimental probabilities for each outcome**:
- For 2: \( \frac{10}{61} \approx 0.164 \)
- For 4: \( \frac{9}{61} \approx 0.148 \)
- For 6: \( \frac{6}{61} \approx 0.098 \)
- For 8: \( \frac{15}{61} \approx 0.246 \)
- For 10: \( \frac{13}{61} \approx 0.213 \)
- For 12: \( \frac{8}{61} \approx 0.131 \)

4. **Calculate the expected probability for each outcome, which is \( \frac{1}{6} \approx 0.167 \)**.

5. **Calculate discrepancies** for each outcome by finding the absolute difference between experimental probabilities and expected probabilities:
- For 2: \( |0.164 - 0.167| \approx 0.003 \)
- For 4: \( |0.148 - 0.167| \approx 0.019 \)
- For 6: \( |0.098 - 0.167| \approx 0.069 \)
- For 8: \( |0.246 - 0.167| \approx 0.079 \)
- For 10: \( |0.213 - 0.167| \approx 0.046 \)
- For 12: \( |0.131 - 0.167| \approx 0.036 \)

6. **Find the smallest discrepancy**:
- The smallest discrepancy is \( 0.003 \) from the outcome \( 2 \).

Thus, the smallest discrepancy between the experimental and expected probabilities in this experiment is:

\[
\boxed{0.003}
\]