Answer :
Certainly! Let's go through each part of the question step by step using the provided table:
### 1. Construct a Frequency Table
First, let's construct the frequency table from the given outcomes and the number of spins:
[tex]\[ \begin{array}{c|c} \text{Outcomes} & \text{Number of Spins} \\ \hline 1 & 13 \\ 2 & 9 \\ 3 & 24 \\ 4 & 7 \\ 5 & 11 \\ 6 & 8 \\ 7 & 10 \\ 8 & 5 \\ 9 & 7 \\ 10 & 6 \\ \end{array} \][/tex]
### 2. Experimental Probability of Spinning a 5
The experimental probability of spinning a 5 is calculated by dividing the number of times 5 was spun by the total number of spins.
[tex]\[ P(5) = \frac{\text{Number of spins of 5}}{\text{Total number of spins}} = \frac{11}{100} = 0.11 \][/tex]
### 3. Experimental Probability of Spinning a 3
The experimental probability of spinning a 3 is calculated similarly:
[tex]\[ P(3) = \frac{\text{Number of spins of 3}}{\text{Total number of spins}} = \frac{24}{100} = 0.24 \][/tex]
### 4. Experimental Probability of Spinning a Number Which is Multiple of 2
A number is a multiple of 2 if it is one of {2, 4, 6, 8, 10}. We sum the spins for these numbers:
[tex]\[ \text{Number of spins of multiples of 2} = 9 + 7 + 8 + 5 + 6 = 35 \][/tex]
The experimental probability is then:
[tex]\[ P(\text{multiple of 2}) = \frac{\text{Number of spins of multiples of 2}}{\text{Total number of spins}} = \frac{35}{100} = 0.35 \][/tex]
### 5. Experimental Probability of Spinning a Number Which is Below 6
A number is below 6 if it is one of {1, 2, 3, 4, 5}. We sum the spins for these numbers:
[tex]\[ \text{Number of spins of numbers below 6} = 13 + 9 + 24 + 7 + 11 = 64 \][/tex]
The experimental probability is then:
[tex]\[ P(\text{below 6}) = \frac{\text{Number of spins of numbers below 6}}{\text{Total number of spins}} = \frac{64}{100} = 0.64 \][/tex]
### 6. Theoretical Probability of Spinning a 2
If the spinner is fair and has 10 sectors, the probability of spinning a specific number (like 2) is:
[tex]\[ P(2) = \frac{1}{10} = 0.1 \][/tex]
Therefore, the detailed solutions for each part are as follows:
1. Frequency Table:
[tex]\[ \begin{array}{c|c} \text{Outcomes} & \text{Number of Spins} \\ \hline 1 & 13 \\ 2 & 9 \\ 3 & 24 \\ 4 & 7 \\ 5 & 11 \\ 6 & 8 \\ 7 & 10 \\ 8 & 5 \\ 9 & 7 \\ 10 & 6 \\ \end{array} \][/tex]
2. Experimental Probability of Spinning a 5: [tex]\(0.11\)[/tex]
3. Experimental Probability of Spinning a 3: [tex]\(0.24\)[/tex]
4. Experimental Probability of Spinning a Number Which is Multiple of 2: [tex]\(0.35\)[/tex]
5. Experimental Probability of Spinning a Number Which is Below 6: [tex]\(0.64\)[/tex]
6. Theoretical Probability of Spinning a 2: [tex]\(0.1\)[/tex]
### 1. Construct a Frequency Table
First, let's construct the frequency table from the given outcomes and the number of spins:
[tex]\[ \begin{array}{c|c} \text{Outcomes} & \text{Number of Spins} \\ \hline 1 & 13 \\ 2 & 9 \\ 3 & 24 \\ 4 & 7 \\ 5 & 11 \\ 6 & 8 \\ 7 & 10 \\ 8 & 5 \\ 9 & 7 \\ 10 & 6 \\ \end{array} \][/tex]
### 2. Experimental Probability of Spinning a 5
The experimental probability of spinning a 5 is calculated by dividing the number of times 5 was spun by the total number of spins.
[tex]\[ P(5) = \frac{\text{Number of spins of 5}}{\text{Total number of spins}} = \frac{11}{100} = 0.11 \][/tex]
### 3. Experimental Probability of Spinning a 3
The experimental probability of spinning a 3 is calculated similarly:
[tex]\[ P(3) = \frac{\text{Number of spins of 3}}{\text{Total number of spins}} = \frac{24}{100} = 0.24 \][/tex]
### 4. Experimental Probability of Spinning a Number Which is Multiple of 2
A number is a multiple of 2 if it is one of {2, 4, 6, 8, 10}. We sum the spins for these numbers:
[tex]\[ \text{Number of spins of multiples of 2} = 9 + 7 + 8 + 5 + 6 = 35 \][/tex]
The experimental probability is then:
[tex]\[ P(\text{multiple of 2}) = \frac{\text{Number of spins of multiples of 2}}{\text{Total number of spins}} = \frac{35}{100} = 0.35 \][/tex]
### 5. Experimental Probability of Spinning a Number Which is Below 6
A number is below 6 if it is one of {1, 2, 3, 4, 5}. We sum the spins for these numbers:
[tex]\[ \text{Number of spins of numbers below 6} = 13 + 9 + 24 + 7 + 11 = 64 \][/tex]
The experimental probability is then:
[tex]\[ P(\text{below 6}) = \frac{\text{Number of spins of numbers below 6}}{\text{Total number of spins}} = \frac{64}{100} = 0.64 \][/tex]
### 6. Theoretical Probability of Spinning a 2
If the spinner is fair and has 10 sectors, the probability of spinning a specific number (like 2) is:
[tex]\[ P(2) = \frac{1}{10} = 0.1 \][/tex]
Therefore, the detailed solutions for each part are as follows:
1. Frequency Table:
[tex]\[ \begin{array}{c|c} \text{Outcomes} & \text{Number of Spins} \\ \hline 1 & 13 \\ 2 & 9 \\ 3 & 24 \\ 4 & 7 \\ 5 & 11 \\ 6 & 8 \\ 7 & 10 \\ 8 & 5 \\ 9 & 7 \\ 10 & 6 \\ \end{array} \][/tex]
2. Experimental Probability of Spinning a 5: [tex]\(0.11\)[/tex]
3. Experimental Probability of Spinning a 3: [tex]\(0.24\)[/tex]
4. Experimental Probability of Spinning a Number Which is Multiple of 2: [tex]\(0.35\)[/tex]
5. Experimental Probability of Spinning a Number Which is Below 6: [tex]\(0.64\)[/tex]
6. Theoretical Probability of Spinning a 2: [tex]\(0.1\)[/tex]
