To determine the expected value of winning on the new "Wheel of Fortune," we can follow these steps:
Step 1: Identify the different amounts of money on the wheel and their respective numbers of slots.
There are:
- \[tex]$200 appears in 6 slots
- \$[/tex]400 appears in 15 slots
- \[tex]$600 appears in 2 slots
- \$[/tex]0 appears in 6 slots
- \[tex]$1000 appears in 1 slot
- \$[/tex]20,000 (the car) appears in 1 slot
Step 2: Calculate the total number of slots on the wheel.
Total slots = [tex]\(6 + 15 + 2 + 6 + 1 + 1 = 31\)[/tex]
Step 3: Determine the probability of landing on each type of slot.
For each amount, the probability is calculated by dividing the number of specific slots by the total number of slots:
- Probability of \[tex]$200: \(\frac{6}{31}\)
- Probability of \$[/tex]400: [tex]\(\frac{15}{31}\)[/tex]
- Probability of \[tex]$600: \(\frac{2}{31}\)
- Probability of \$[/tex]0: [tex]\(\frac{6}{31}\)[/tex]
- Probability of \[tex]$1000: \(\frac{1}{31}\)
- Probability of \$[/tex]20,000: [tex]\(\frac{1}{31}\)[/tex]
Step 4: Calculate the expected value.
The expected value [tex]\(E\)[/tex] is found by multiplying each monetary amount by its respective probability and summing the results:
[tex]\[
E = \left( 200 \times \frac{6}{31} \right) + \left( 400 \times \frac{15}{31} \right) + \left( 600 \times \frac{2}{31} \right) + \left( 0 \times \frac{6}{31} \right) + \left( 1000 \times \frac{1}{31} \right) + \left( 20000 \times \frac{1}{31} \right)
\][/tex]
[tex]\[
E = \left( \frac{1200}{31} \right) + \left( \frac{6000}{31} \right) + \left( \frac{1200}{31} \right) + \left( 0 \right) + \left( \frac{1000}{31} \right) + \left( \frac{20000}{31} \right)
\][/tex]
[tex]\[
E = \frac{1200 + 6000 + 1200 + 0 + 1000 + 20000}{31}
\][/tex]
[tex]\[
E = \frac{29200}{31}
\][/tex]
[tex]\[
E \approx 948.3870967741935
\][/tex]
Therefore, the expected value of winning on the new "Wheel of Fortune" is approximately \$948.39.
