Answer :
Let's go through the problem step-by-step to find the required probabilities.
1. Identify Total Rolls:
The die is rolled 15 times, which means the total number of rolls is 15.
2. Identify Odd Numbers:
The numbers on the die that are considered odd are 1, 3, 5, and 7.
3. Count Odd Results:
From the given results: [3, 4, 5, 2, 7, 1, 3, 7, 2, 6, 2, 1, 7, 3, 6], we need to count how many times an odd number appears.
- 3 appears 3 times.
- 5 appears 1 time.
- 7 appears 3 times.
- 1 appears 2 times.
Thus, the total count of odd results is [tex]\(3 + 1 + 3 + 2 = 9\)[/tex].
4. Calculate Empirical Probability:
The empirical probability is the ratio of the number of odd results to the total number of rolls, expressed as a percentage:
[tex]\[ \text{Empirical Probability} = \left(\frac{\text{Number of Odd Results}}{\text{Total Rolls}}\right) \times 100 = \left(\frac{9}{15}\right) \times 100 = 60\% \][/tex]
5. Calculate Theoretical Probability:
The theoretical probability of rolling an odd number (1, 3, 5, 7) on an 8-sided die is:
[tex]\[ \text{Theoretical Probability} = \left(\frac{\text{Number of Odd Sides}}{\text{Total Sides}}\right) \times 100 = \left(\frac{4}{8}\right) \times 100 = 50\% \][/tex]
6. Calculate the Difference:
The difference between the empirical and theoretical probabilities is:
[tex]\[ \text{Difference} = \text{Empirical Probability} - \text{Theoretical Probability} = 60\% - 50\% = 10\% \][/tex]
7. Final Answers for the Boxes:
The empirical probability of rolling an odd number is [tex]\(60\%\)[/tex], which is [tex]\(10\%\)[/tex] more than the theoretical probability.
Therefore, you should fill in the boxes as follows:
The empirical probability of rolling an odd number is [tex]\(\boxed{60}\)[/tex] %, which is [tex]\(\boxed{10}\)[/tex] % more than the theoretical probability.
1. Identify Total Rolls:
The die is rolled 15 times, which means the total number of rolls is 15.
2. Identify Odd Numbers:
The numbers on the die that are considered odd are 1, 3, 5, and 7.
3. Count Odd Results:
From the given results: [3, 4, 5, 2, 7, 1, 3, 7, 2, 6, 2, 1, 7, 3, 6], we need to count how many times an odd number appears.
- 3 appears 3 times.
- 5 appears 1 time.
- 7 appears 3 times.
- 1 appears 2 times.
Thus, the total count of odd results is [tex]\(3 + 1 + 3 + 2 = 9\)[/tex].
4. Calculate Empirical Probability:
The empirical probability is the ratio of the number of odd results to the total number of rolls, expressed as a percentage:
[tex]\[ \text{Empirical Probability} = \left(\frac{\text{Number of Odd Results}}{\text{Total Rolls}}\right) \times 100 = \left(\frac{9}{15}\right) \times 100 = 60\% \][/tex]
5. Calculate Theoretical Probability:
The theoretical probability of rolling an odd number (1, 3, 5, 7) on an 8-sided die is:
[tex]\[ \text{Theoretical Probability} = \left(\frac{\text{Number of Odd Sides}}{\text{Total Sides}}\right) \times 100 = \left(\frac{4}{8}\right) \times 100 = 50\% \][/tex]
6. Calculate the Difference:
The difference between the empirical and theoretical probabilities is:
[tex]\[ \text{Difference} = \text{Empirical Probability} - \text{Theoretical Probability} = 60\% - 50\% = 10\% \][/tex]
7. Final Answers for the Boxes:
The empirical probability of rolling an odd number is [tex]\(60\%\)[/tex], which is [tex]\(10\%\)[/tex] more than the theoretical probability.
Therefore, you should fill in the boxes as follows:
The empirical probability of rolling an odd number is [tex]\(\boxed{60}\)[/tex] %, which is [tex]\(\boxed{10}\)[/tex] % more than the theoretical probability.
