Let's break down the problem step-by-step to calculate the expected value and determine the price that would make the game fair.

1. Understanding the game:

• A player draws a card from a standard 52-card deck.
• Cost to play: $3 (this will be subtracted from the winnings).
• If a heart is drawn (13 hearts in the deck):
  • The player wins $6.
  • Their net gain is: $6 - $3 = $3.
• If a non-heart card is drawn (39 non-heart cards):
  • The player loses the $3 fee.
  • Their net loss is: $0 - $3 = -$3.

2. Probability of drawing a heart or a non-heart card:

• Probability of drawing a heart:
  $P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$
• Probability of drawing a non-heart:
  $P(\text{non-heart}) = \frac{39}{52} = \frac{3}{4}$

3. Expected value calculation:

The expected value (EV) is the sum of the possible outcomes, each weighted by their respective probabilities. We calculate it by multiplying the net outcome by the probability for each scenario:

$EV = \left( \frac{1}{4} \times 3 \right) + \left( \frac{3}{4} \times (-3) \right)$

Let's compute this:

$EV = \left( \frac{3}{4} \right) + \left( -\frac{9}{4} \right) = \frac{3}{4} - \frac{9}{4} = \frac{-6}{4} = -1.5$

4. Conclusion:

• The expected value is -1.5. This means, on average, the player loses $1.50 for every game played.

5. Fair price:

For the game to be fair, the expected value should be 0. To make this happen, we need to adjust the winnings when a heart is drawn. Let's denote the fair prize as $X$.

To make the game fair, the expected value must be 0:

$0 = \left( \frac{1}{4} \times (X - 3) \right) + \left( \frac{3}{4} \times (-3) \right)$

Simplifying:

$0 = \frac{1}{4} \times (X - 3) - \frac{9}{4}$

Multiply the entire equation by 4 to eliminate fractions:

$0 = (X - 3) - 9$

Solve for $X$:

$X - 3 = 9$

$X = 12$

Final Answer:

To make the game fair, the player should win $12 when a heart is drawn.

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Would you like further clarification or additional details on any step?

Related Questions:

1. What is the variance of this game based on the current probabilities?
2. How does increasing the prize for drawing a heart affect the expected value?
3. What would the expected value be if the cost of playing the game changed to $4?
4. How does the calculation change if jokers are added to the deck?
5. What if the number of hearts in the deck was different, say 10 instead of 13?

Tip:

Expected value helps in evaluating whether a game is fair and how the probability of outcomes influences your chances of winning or losing over the long run.