Let's denote the following:
- Points won: 20 points, 25 points, 30 points
- Their probabilities: 0.6, [tex]\( x \)[/tex], [tex]\( y \)[/tex]
We are given that the expected number of points is 22.5. Therefore, we can set up the equation for the expected value as follows:
[tex]\[ \text{Expected value} = (20 \times 0.6) + (25 \times x) + (30 \times y) = 22.5 \][/tex]
We are also given that the sum of all probabilities must equal 1:
[tex]\[ 0.6 + x + y = 1 \][/tex]
First, let's solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 1 - 0.6 - x \][/tex]
[tex]\[ y = 0.4 - x \][/tex]
Next, substitute [tex]\( y \)[/tex] into the expected value equation:
[tex]\[ 20 \times 0.6 + 25 \times x + 30 \times (0.4 - x) = 22.5 \][/tex]
Calculate [tex]\( 20 \times 0.6 \)[/tex]:
[tex]\[ 12 \][/tex]
Substitute the value into the equation:
[tex]\[ 12 + 25x + 30(0.4 - x) = 22.5 \][/tex]
Distribute the 30:
[tex]\[ 12 + 25x + 12 - 30x = 22.5 \][/tex]
Combine like terms:
[tex]\[ 24 - 5x = 22.5 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ -5x = 22.5 - 24 \][/tex]
[tex]\[ -5x = -1.5 \][/tex]
[tex]\[ x = 0.3 \][/tex]
Now, substitute the value of [tex]\( x \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[ y = 0.4 - 0.3 \][/tex]
[tex]\[ y = 0.1 \][/tex]
The values are:
[tex]\[ x = 0.3 \][/tex]
[tex]\[ y = 0.1 \][/tex]
