To determine the geometric probability of scoring an odd number of points on Juan's game board, we need to follow these steps:
### Step 1: Understand the Game Board
First, we need to understand the layout of the game board. The board is a 5x5 matrix as follows:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
5 & 1 & 2 & 1 & 5 \\ \hline
1 & 4 & 4 & 4 & 1 \\ \hline
2 & 4 & 5 & 4 & 2 \\ \hline
1 & 4 & 4 & 4 & 1 \\ \hline
5 & 1 & 2 & 1 & 5 \\ \hline
\end{array}
\][/tex]
### Step 2: Count the Odd and Even Numbers
To calculate the probability of landing on an odd number, we first count the odd numbers on the board:
Odd number counts:
- The 1st row has the numbers: 5, 1, 2, 1, 5 -> 4 odd numbers.
- The 2nd row has the numbers: 1, 4, 4, 4, 1 -> 2 odd numbers.
- The 3rd row has the numbers: 2, 4, 5, 4, 2 -> 1 odd number.
- The 4th row has the numbers: 1, 4, 4, 4, 1 -> 2 odd numbers.
- The 5th row has the numbers: 5, 1, 2, 1, 5 -> 4 odd numbers.
So, the total count of odd numbers is:
[tex]\[
4 + 2 + 1 + 2 + 4 = 13
\][/tex]
### Step 3: Count the Total Numbers
Next, calculate the total number of cells in the game board. Since it's a 5x5 grid:
[tex]\[
5 \times 5 = 25
\][/tex]
### Step 4: Calculate the Probability
We calculate the probability of scoring on an odd number by dividing the number of odd numbers by the total number of cells:
[tex]\[
\text{Probability of odd number} = \frac{\text{Number of odd numbers}}{\text{Total number of cells}} = \frac{13}{25}
\][/tex]
### Step 5: Simplify and Convert to Decimal
Perform the division:
[tex]\[
\frac{13}{25} = 0.52
\][/tex]
### Conclusion
The geometric probability of scoring an odd number of points on Juan's game board is [tex]\( \boxed{0.52} \)[/tex].
