Answer :
Certainly! Let's analyze this problem step-by-step based on the given observed frequencies and the statements provided.
### Given Observed Frequencies:
[tex]\[ \begin{array}{|c|c|} \hline Number & \text{Observed Frequency} \\ \hline 1 & 0 \\ \hline 2 & 0 \\ \hline 3 & 0 \\ \hline 4 & 0 \\ \hline 5 & 0 \\ \hline 6 & 0 \\ \hline \text{Total} & 0 \\ \hline \end{array} \][/tex]
### Total Rolls:
The total number of rolls is the sum of the observed frequencies:
[tex]\[ 0 + 0 + 0 + 0 + 0 + 0 = 0 \][/tex]
So, the total number of rolls is 0.
### Analysis of Statements:
1. As the number of trials increases, experimental probability is closer to theoretical probability.
- The experimental probability converges to the theoretical probability as the number of trials increases. This is a fundamental principle in probability theory due to the Law of Large Numbers.
2. As the number of trials increases, theoretical probability changes to more closely match the experimental probability.
- Theoretical probability is constant and does not depend on the number of trials. It is an inherent property based on the possible outcomes of the random experiment.
3. As the number of trials increases, there is no change in the experimental probabilities.
- The experimental probabilities can vary with additional trials; they become more stable and closer to the theoretical probabilities but are not constant unless observed over an infinite number of trials.
4. As the number of trials increases, there is no change in the theoretical probabilities.
- The theoretical probabilities are static and do not change regardless of the number of trials.
### Determining the True Statements:
Based on our analysis, let's identify the true statements:
- Statement 1 is true: This is because the experimental probability converges to the theoretical probability with an increasing number of trials.
- Statement 4 is true: Theoretical probabilities remain constant regardless of the number of trials.
Hence, the true statements are:
[tex]\[ [1, 4] \][/tex]
Therefore, given the total rolls of 0, the true statements are:
### Result:
[tex]\[ (0, [4]) \][/tex]
So, the total number of rolls is 0, and the correct statements based on the given observed frequencies are:
- As the number of trials increases, there is no change in the theoretical probabilities.
### Given Observed Frequencies:
[tex]\[ \begin{array}{|c|c|} \hline Number & \text{Observed Frequency} \\ \hline 1 & 0 \\ \hline 2 & 0 \\ \hline 3 & 0 \\ \hline 4 & 0 \\ \hline 5 & 0 \\ \hline 6 & 0 \\ \hline \text{Total} & 0 \\ \hline \end{array} \][/tex]
### Total Rolls:
The total number of rolls is the sum of the observed frequencies:
[tex]\[ 0 + 0 + 0 + 0 + 0 + 0 = 0 \][/tex]
So, the total number of rolls is 0.
### Analysis of Statements:
1. As the number of trials increases, experimental probability is closer to theoretical probability.
- The experimental probability converges to the theoretical probability as the number of trials increases. This is a fundamental principle in probability theory due to the Law of Large Numbers.
2. As the number of trials increases, theoretical probability changes to more closely match the experimental probability.
- Theoretical probability is constant and does not depend on the number of trials. It is an inherent property based on the possible outcomes of the random experiment.
3. As the number of trials increases, there is no change in the experimental probabilities.
- The experimental probabilities can vary with additional trials; they become more stable and closer to the theoretical probabilities but are not constant unless observed over an infinite number of trials.
4. As the number of trials increases, there is no change in the theoretical probabilities.
- The theoretical probabilities are static and do not change regardless of the number of trials.
### Determining the True Statements:
Based on our analysis, let's identify the true statements:
- Statement 1 is true: This is because the experimental probability converges to the theoretical probability with an increasing number of trials.
- Statement 4 is true: Theoretical probabilities remain constant regardless of the number of trials.
Hence, the true statements are:
[tex]\[ [1, 4] \][/tex]
Therefore, given the total rolls of 0, the true statements are:
### Result:
[tex]\[ (0, [4]) \][/tex]
So, the total number of rolls is 0, and the correct statements based on the given observed frequencies are:
- As the number of trials increases, there is no change in the theoretical probabilities.
