to play the game, which must be subtracted from the winnings. If a club is drawn, the player wins 21 dollars; otherwis they lose their 14 dollars. Calculate the price that would make the game fair. The expected value of this game is dollars. (Round to the nearest cent as needed.)
to play the game, which must be subtracted from the winnings. If a club is drawn, the player wins 21 dollars; otherwis they lose their 14 dollars. Calculate the price that would make the game fair. The expected value of this game is dollars. (Round to the nearest cent as needed.)
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![### Calculating the Expected Value of a Card Game
In this exercise, we will determine the expected value for a game involving a standard 52-card deck. This will help us understand how much a player can expect to win or lose on average per game.
#### Game Rules:
1. A player must pay $14 to participate in the game.
2. The goal is to draw a card from a standard 52-card deck.
3. If a club is drawn (one of the 13 clubs in the deck), the player wins $21.
4. If any other suit is drawn (one of the 39 non-club cards), the player loses their $14.
#### Problem Statement:
Calculate the price that would make the game fair for the player.
#### Calculations:
1. **Determine the Probability of Each Outcome:**
- Probability of drawing a club (P(club)):
\[
\frac{13}{52} = \frac{1}{4} = 0.25
\]
- Probability of drawing a non-club (P(non-club)):
\[
\frac{39}{52} = \frac{3}{4} = 0.75
\]
2. **Calculate the Winnings and Losses:**
- Winning amount when a club is drawn: $21
- Loss amount when a non-club is drawn: $14
3. **Expected Value Calculation:**
The expected value (EV) is given by the formula:
\[
\text{EV} = (\text{P(club)} \times \text{Winnings}) + (\text{P(non-club)} \times \text{Loss})
\]
Substituting the probabilities and amounts:
\[
\text{EV} = (0.25 \times 21) + (0.75 \times (-14))
\]
\[
\text{EV} = (5.25) + (-10.5)
\]
\[
\text{EV} = -5.25
\]
4. **Fair Game Calculation:**
For the game to be fair, the expected value should be zero. Let "X" be the price that makes the game fair:
\[
0 = (0.25 \times 21) + (0.75 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F128593f5-7ea4-4e3b-94cd-55abf91c7165%2Ffbe59ff9-f48b-42c5-ba13-2c086f380bb0%2Far6rj0f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Expected Value of a Card Game
In this exercise, we will determine the expected value for a game involving a standard 52-card deck. This will help us understand how much a player can expect to win or lose on average per game.
#### Game Rules:
1. A player must pay $14 to participate in the game.
2. The goal is to draw a card from a standard 52-card deck.
3. If a club is drawn (one of the 13 clubs in the deck), the player wins $21.
4. If any other suit is drawn (one of the 39 non-club cards), the player loses their $14.
#### Problem Statement:
Calculate the price that would make the game fair for the player.
#### Calculations:
1. **Determine the Probability of Each Outcome:**
- Probability of drawing a club (P(club)):
\[
\frac{13}{52} = \frac{1}{4} = 0.25
\]
- Probability of drawing a non-club (P(non-club)):
\[
\frac{39}{52} = \frac{3}{4} = 0.75
\]
2. **Calculate the Winnings and Losses:**
- Winning amount when a club is drawn: $21
- Loss amount when a non-club is drawn: $14
3. **Expected Value Calculation:**
The expected value (EV) is given by the formula:
\[
\text{EV} = (\text{P(club)} \times \text{Winnings}) + (\text{P(non-club)} \times \text{Loss})
\]
Substituting the probabilities and amounts:
\[
\text{EV} = (0.25 \times 21) + (0.75 \times (-14))
\]
\[
\text{EV} = (5.25) + (-10.5)
\]
\[
\text{EV} = -5.25
\]
4. **Fair Game Calculation:**
For the game to be fair, the expected value should be zero. Let "X" be the price that makes the game fair:
\[
0 = (0.25 \times 21) + (0.75 \
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