To address the various questions about Nathan's number cube rolls, let's investigate the situation step-by-step.
1. Frequencies of Each Number Rolled:
- 1: 11 times
- 2: 16 times
- 3: 14 times
- 4: 20 times
- 5: 12 times
- 6: 17 times
2. Total Number of Rolls:
- The total number of rolls can be calculated by summing the frequency of each number.
[tex]\[
\text{Total rolls} = 11 + 16 + 14 + 20 + 12 + 17 = 90
\][/tex]
3. Relative Frequency of Rolling a 4:
- The relative frequency is the ratio of the number of times the event occurs to the total number of trials.
[tex]\[
\text{Relative frequency of rolling a 4} = \frac{\text{Frequency of 4}}{\text{Total rolls}} = \frac{20}{90} = \frac{2}{9} \approx 0.2222
\][/tex]
4. Experimental Probability of Rolling a 3:
- The experimental probability is the ratio of the number of times the specific outcome occurs to the total number of trials.
[tex]\[
\text{Experimental probability of rolling a 3} = \frac{\text{Frequency of 3}}{\text{Total rolls}} = \frac{14}{90} \approx 0.1556
\][/tex]
5. Theoretical Probability of Rolling Any Specific Number on a Fair Die:
- Since the number cube is fair, each outcome has an equal chance of occurring.
[tex]\[
\text{Theoretical probability of rolling a 3} = \frac{1}{6} \approx 0.1667
\][/tex]
6. Comparison of Statements:
- Relative Frequency of Rolling a 4 is [tex]\( \frac{2}{9} \)[/tex]:
[tex]\[
\frac{20}{90} = \frac{2}{9} \text{, which is true.}
\][/tex]
- Experimental Probability of Rolling a 3 is Greater than the Theoretical Probability of Rolling a 3:
[tex]\[
0.1556 \not> 0.1667 \text{, which is false.}
\][/tex]
Considering the detailed mathematical process and the results obtained, the correct statements related to this situation are:
1. The relative frequency of rolling a 4 is [tex]\( \frac{2}{9} \)[/tex].
The other statement mentioned (Experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3) is false. Therefore, only one of the given options accurately represents the situation.
