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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.

You are playing a game where you draw a card from a standard deck, and you will win $5 if you draw a face card, $4 if you draw an ace, and lose $6 if you draw any other card. What is the expected gain from this game?

Learn how to calculate the expected gain from a card-drawing game involving a standard deck. Understand the use of probability and expected value formulas to determine outcomes based on drawing face cards, aces, or other cards. Ideal for students in Grades 10-12.

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