To calculate the expected value of the game, we first identify the possible outcomes, their probabilities, and their respective payouts.

Step 1: Define the outcomes and probabilities

1. Aces: There are 4 aces in a standard deck of 52 cards.
   Probability of drawing an ace = $\frac{4}{52} = \frac{1}{13}$.

2. Face cards: There are 12 face cards (Jack, Queen, King in each of the four suits).
   Probability of drawing a face card = $\frac{12}{52} = \frac{3}{13}$.

3. Numbered cards: There are 36 numbered cards (2 through 10 in each suit).
   Probability of drawing a numbered card = $\frac{36}{52} = \frac{9}{13}$.

Step 2: Calculate the net payout for each outcome

The player pays $5.00 to play, so the net winnings are:

• Ace: $18 - 5 = 13$.
• Face card: $5 - 5 = 0$.
• Numbered card: $2 - 5 = -3$.

Step 3: Compute the expected value

The expected value $E(X)$ is the sum of the products of each outcome's net payout and its probability:

$E(X) = (P(\text{Ace}) \cdot \text{Net Payout for Ace}) + (P(\text{Face card}) \cdot \text{Net Payout for Face card}) + (P(\text{Numbered card}) \cdot \text{Net Payout for Numbered card})$

Substituting the values: $E(X) = \left(\frac{1}{13} \cdot 13\right) + \left(\frac{3}{13} \cdot 0\right) + \left(\frac{9}{13} \cdot (-3)\right)$

Simplify: $E(X) = \frac{13}{13} + \frac{0}{13} - \frac{27}{13}$

$E(X) = 1 - \frac{27}{13}$

$E(X) = \frac{13}{13} - \frac{27}{13} = \frac{-14}{13} \approx -1.08$

Step 4: Interpret the result

The expected value of the game is approximately -1.08. This means that, on average, the player will lose $1.08 per game.

---

Would you like more details or have further questions?
Here are some related questions:

1. How does the house profit from this game?
2. How would the expected value change if the payouts were increased?
3. What is the probability of the player breaking even or winning?
4. Can you simulate multiple rounds of this game to observe the average outcomes?
5. How would the expected value change if the cost to play was reduced?

Tip: Always check the game's expected value before deciding whether to play—it reveals whether the odds are in your favor!