Question

for the wheel pictured on the right, assume that a person spins the pointer and is awarded the amount indicated by the pointer. if it costs \\$5 to play the game, determine (a) the expectation of a person who plays the game and (b) the fair price to play the game.

a. what is the expectation? \\$
(round to the nearest cent as needed.)

b. what is the fair price to play the game? \\$
(round to the nearest cent as needed.)

Explanation:

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"concepts_used": [
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<reasoning>

Determine the probabilities of the spinner outcomes

Using the Spinner Probability knowledge point
\[

$$\begin{aligned} P(\$2) &= 0.5 \\ P(\$11) &= 0.5 \end{aligned}$$

\]

Calculate the expected gross winnings

Using the Expected Value knowledge point
\[

$$\begin{aligned} E_{\text{gross}} &= (\$2 \times 0.5) + (\$11 \times 0.5) \\ E_{\text{gross}} &= \$1.00 + \$5.50 = \$6.50 \end{aligned}$$

\]

Calculate the expectation of a player

Using the Expected Value knowledge point
\[

$$\begin{aligned} E_{\text{net}} &= E_{\text{gross}} - \text{Cost to play} \\ E_{\text{net}} &= \$6.50 - \$5.00 = \$1.50 \end{aligned}$$

\]

Determine the fair price to play the game

Using the Fair Price knowledge point
\[

$$\begin{aligned} \text{Fair Price} &= E_{\text{gross}} \\ \text{Fair Price} &= \$6.50 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

What is the expectation? $<blank>1.50</blank>
(Round to the nearest cent as needed.)

Question b

What is the fair price to play the game? $<blank>6.50</blank>
(Round to the nearest cent as needed.)
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Determine the probabilities of the spinner outcomes

Using the Spinner Probability knowledge point
\[

$$\begin{aligned} P(\$2) &= 0.5 \\ P(\$11) &= 0.5 \end{aligned}$$

\]

Calculate the expected gross winnings

Using the Expected Value knowledge point
\[

$$\begin{aligned} E_{\text{gross}} &= (\$2 \times 0.5) + (\$11 \times 0.5) \\ E_{\text{gross}} &= \$1.00 + \$5.50 = \$6.50 \end{aligned}$$

\]

Calculate the expectation of a player

Using the Expected Value knowledge point
\[

$$\begin{aligned} E_{\text{net}} &= E_{\text{gross}} - \text{Cost to play} \\ E_{\text{net}} &= \$6.50 - \$5.00 = \$1.50 \end{aligned}$$

\]

Determine the fair price to play the game

Using the Fair Price knowledge point
\[

$$\begin{aligned} \text{Fair Price} &= E_{\text{gross}} \\ \text{Fair Price} &= \$6.50 \end{aligned}$$

\]
</reasoning>

<answer>

Question a

What is the expectation? $<blank>1.50</blank>
(Round to the nearest cent as needed.)

Question b

What is the fair price to play the game? $<blank>6.50</blank>
(Round to the nearest cent as needed.)
</answer>

<post_analysis>
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