1. **State the problem:** Calculate the expected value of a game where a card is drawn from a standard 52-card deck. 2. **Identify the outcomes and their probabilities:** - Black face cards: There are 3 face cards (Jack, Queen, King) in each black suit (clubs and spades), so $3 \times 2 = 6$ cards. - Red Aces: There are 2 red Aces (hearts and diamonds). - Other cards: Total cards $52$ minus black face cards $6$ minus red Aces $2$ equals $44$ cards. 3. **Calculate probabilities:** - $P(\text{black face card}) = \frac{6}{52}$ - $P(\text{red Ace}) = \frac{2}{52}$ - $P(\text{other card}) = \frac{44}{52}$ 4. **Calculate expected winnings:** $$E(\text{winnings}) = 40 \times \frac{6}{52} + 15 \times \frac{2}{52} + 8 \times \frac{44}{52}$$ 5. **Calculate each term:** $$40 \times \frac{6}{52} = \frac{240}{52}$$ $$15 \times \frac{2}{52} = \frac{30}{52}$$ $$8 \times \frac{44}{52} = \frac{352}{52}$$ 6. **Sum the expected winnings:** $$E(\text{winnings}) = \frac{240}{52} + \frac{30}{52} + \frac{352}{52} = \frac{622}{52}$$ 7. **Simplify the fraction:** $$\frac{622}{52} = \frac{\cancel{2}311}{\cancel{2}26}$$ 8. **Calculate decimal value:** $$\frac{311}{26} \approx 11.96$$ 9. **Subtract the cost to play ($10$):** $$E(\text{net}) = 11.96 - 10 = 1.96$$ **Final answer:** The expected value of the game is approximately $1.96$ per play, meaning on average the player gains $1.96$ per game.