1. **Problem statement:** We have a list of drawn numbers from an urn: 3, 4, 5, 6. We need to find the absolute frequencies, relative frequencies (as fractions and decimals), and expected relative frequencies for 1000 trials. 2. **Step a) Absolute frequencies:** Count each number's occurrences. - Count of 3: 20 - Count of 4: 18 - Count of 5: 14 - Count of 6: 6 3. **Step b) Relative frequencies:** Total draws $n = 20 + 18 + 14 + 6 = 58$. - Relative frequency for 3: $\frac{20}{58}$ - Relative frequency for 4: $\frac{18}{58}$ - Relative frequency for 5: $\frac{14}{58}$ - Relative frequency for 6: $\frac{6}{58}$ Simplify fractions where possible: - $\frac{20}{58} = \frac{10}{29}$ - $\frac{18}{58} = \frac{9}{29}$ - $\frac{14}{58} = \frac{7}{29}$ - $\frac{6}{58} = \frac{3}{29}$ Decimal equivalents: - $\frac{10}{29} \approx 0.3448$ - $\frac{9}{29} \approx 0.3103$ - $\frac{7}{29} \approx 0.2414$ - $\frac{3}{29} \approx 0.1034$ Sum of relative frequencies as fraction: $$\frac{10}{29} + \frac{9}{29} + \frac{7}{29} + \frac{3}{29} = \frac{29}{29} = 1$$ Sum as decimal: $0.3448 + 0.3103 + 0.2414 + 0.1034 = 1.000$ (approx.) 4. **Step c) Expected relative frequencies for 1000 trials:** - For 3: $0.3448 \times 1000 \approx 345$ - For 4: $0.3103 \times 1000 \approx 310$ - For 5: $0.2414 \times 1000 \approx 241$ - For 6: $0.1034 \times 1000 \approx 104$ **Reasoning:** The expected relative frequencies are based on the observed relative frequencies from the sample, assuming the experiment is repeated many times and the probabilities remain stable. --- **Summary Table:** | Ergebnis | Absolute Häufigkeit | Relative Häufigkeit (Bruch) | Relative Häufigkeit (Dezimal) | |----------|---------------------|-----------------------------|-------------------------------| | 3 | 20 | $\frac{10}{29}$ | 0.3448 | | 4 | 18 | $\frac{9}{29}$ | 0.3103 | | 5 | 14 | $\frac{7}{29}$ | 0.2414 | | 6 | 6 | $\frac{3}{29}$ | 0.1034 | | Summe | 58 | 1 | 1.0000 |