Let's solve this problem step by step.

The problem asks for the expected gain from a game where:

• You win $5 if you draw a face card,
• You win $4 if you draw an ace,
• You lose $6 if you draw any other card.

Understanding the deck:

• A standard deck of cards has 52 cards.
• There are 12 face cards (4 Jacks, 4 Queens, 4 Kings).
• There are 4 aces.
• The remaining 36 cards are non-face and non-ace cards.

Assigning probabilities:

• Probability of drawing a face card: $P(\text{face card}) = \frac{12}{52}$
• Probability of drawing an ace: $P(\text{ace}) = \frac{4}{52}$
• Probability of drawing any other card: $P(\text{other}) = \frac{36}{52}$

Calculating expected gain:

The expected value (E) is calculated using the formula:

$E = (\text{gain from face card} \times P(\text{face card})) + (\text{gain from ace} \times P(\text{ace})) + (\text{loss from other cards} \times P(\text{other}))$

Substituting the values: $E = (5 \times \frac{12}{52}) + (4 \times \frac{4}{52}) + (-6 \times \frac{36}{52})$

Now, calculating each term:

• $5 \times \frac{12}{52} = \frac{60}{52} \approx 1.15$
• $4 \times \frac{4}{52} = \frac{16}{52} \approx 0.31$
• $-6 \times \frac{36}{52} = \frac{-216}{52} \approx -4.15$

Adding these together:

$E \approx 1.15 + 0.31 - 4.15 = -2.69$

The expected gain from the game is approximately -2.69. This means that, on average, you lose $2.69 per game.

Do you want further clarification or details on this calculation?

Here are 5 related questions you can try:

1. What is the expected value if the loss for any other card is changed to $5?
2. How does the expected value change if you add another face card?
3. What is the probability of not losing any money?
4. If you win $6 for an ace instead of $4, what will be the new expected value?
5. How would the calculation change if the deck had jokers?

Tip: Always break down the problem into probabilities and corresponding outcomes before computing the expected value.