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6. "Wheel of Fortune" just got a new wheel! On it, there are 6 slots worth \[tex]$200, 15 slots worth \$[/tex]400, 2 slots worth \[tex]$600, 6 slots with no money, 1 slot with \$[/tex]1000, and 1 slot with a car worth \[tex]$20,000. What is the expected value of winning?

\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Amount & \$[/tex]200 & \[tex]$400 & \$[/tex]600 & \[tex]$0 & \$[/tex]1000 & \$20,000 (car) \\
\hline
Probability & \frac{6}{31} & \frac{15}{31} & \frac{2}{31} & \frac{6}{31} & \frac{1}{31} & \frac{1}{31} \\
\hline
\end{tabular}
\]

Sagot :

GinnyAnswer
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.
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