Answer :
Sure, I will guide you through the detailed solution to each part of the problem.
### Part a: Complete the Table to Show the Possible Scores
Letâs construct the score table for two spinners, each having outcomes from 1 to 6.
- If the arrows on both spinners land on the same number, the score is 0.
- If the arrows land on different numbers, the score is the sum of those numbers.
We will fill the table row by row.
#### Spinner Scores Table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 0 & 3 & 4 & 5 & 6 & 7 \\ \hline 2 & 3 & 0 & 5 & 6 & 7 & 8 \\ \hline 3 & 4 & 5 & 0 & 7 & 8 & 9 \\ \hline 4 & 5 & 6 & 7 & 0 & 9 & 10 \\ \hline 5 & 6 & 7 & 8 & 9 & 0 & 11 \\ \hline 6 & 7 & 8 & 9 & 10 & 11 & 0 \\ \hline \end{array} \][/tex]
### Part b: What is the Probability that the Score is an Odd Number?
To find this probability, we need to determine the total number of possible outcomes and count the outcomes that result in an odd score.
#### Step-by-Step Calculation:
1. Total Possible Outcomes:
There are 6 outcomes for each spinner. So, the total number of outcomes for both spinners = \(6 \times 6 = 36\).
2. List of Scores:
Flattening the score table, the possible scores are:
[tex]\[0, 3, 4, 5, 6, 7, 3, 0, 5, 6, 7, 8, 4, 5, 0, 7, 8, 9, 5, 6, 7, 0, 9, 10, 6, 7, 8, 9, 0, 11, 7, 8, 9, 10, 11, 0\][/tex]
3. Count of Odd Scores:
The odd scores from the list are: 3, 5, 7, 5, 7, 9, 5, 7, 9, 7, 9, 11, 7, 9, 11.
Counting these odd scores, we have 18 odd scores.
4. Probability of Odd Score:
[tex]\[ \text{Probability} = \frac{\text{Number of Odd Scores}}{\text{Total Scores}} = \frac{18}{36} = 0.5 \][/tex]
So, the probability that the score is an odd number is [tex]\(\boxed{0.5}\)[/tex].
### Part a: Complete the Table to Show the Possible Scores
Letâs construct the score table for two spinners, each having outcomes from 1 to 6.
- If the arrows on both spinners land on the same number, the score is 0.
- If the arrows land on different numbers, the score is the sum of those numbers.
We will fill the table row by row.
#### Spinner Scores Table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 0 & 3 & 4 & 5 & 6 & 7 \\ \hline 2 & 3 & 0 & 5 & 6 & 7 & 8 \\ \hline 3 & 4 & 5 & 0 & 7 & 8 & 9 \\ \hline 4 & 5 & 6 & 7 & 0 & 9 & 10 \\ \hline 5 & 6 & 7 & 8 & 9 & 0 & 11 \\ \hline 6 & 7 & 8 & 9 & 10 & 11 & 0 \\ \hline \end{array} \][/tex]
### Part b: What is the Probability that the Score is an Odd Number?
To find this probability, we need to determine the total number of possible outcomes and count the outcomes that result in an odd score.
#### Step-by-Step Calculation:
1. Total Possible Outcomes:
There are 6 outcomes for each spinner. So, the total number of outcomes for both spinners = \(6 \times 6 = 36\).
2. List of Scores:
Flattening the score table, the possible scores are:
[tex]\[0, 3, 4, 5, 6, 7, 3, 0, 5, 6, 7, 8, 4, 5, 0, 7, 8, 9, 5, 6, 7, 0, 9, 10, 6, 7, 8, 9, 0, 11, 7, 8, 9, 10, 11, 0\][/tex]
3. Count of Odd Scores:
The odd scores from the list are: 3, 5, 7, 5, 7, 9, 5, 7, 9, 7, 9, 11, 7, 9, 11.
Counting these odd scores, we have 18 odd scores.
4. Probability of Odd Score:
[tex]\[ \text{Probability} = \frac{\text{Number of Odd Scores}}{\text{Total Scores}} = \frac{18}{36} = 0.5 \][/tex]
So, the probability that the score is an odd number is [tex]\(\boxed{0.5}\)[/tex].
