To find \( P(\text{ace} | \text{red}) \), we can use Bayes’ Theorem or consider the problem directly with conditional probability.
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Identify the total number of cards: A standard deck has 52 cards.
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Identify the relevant subsets:
- There are 4 aces in a deck.
- There are 26 red cards (13 hearts and 13 diamonds).
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Calculate \( P(\text{red}) \): The probability of selecting a red card is: \[ P(\text{red}) = \frac{26}{52} = \frac{1}{2} \]
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Calculate \( P(\text{ace} \cap \text{red}) \): The probability of selecting an ace that is also red (the red aces) is: \[ P(\text{ace} \cap \text{red}) = \frac{2}{52} = \frac{1}{26} \]
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Now, use the formula for conditional probability: \[ P(\text{ace} | \text{red}) = \frac{P(\text{ace} \cap \text{red})}{P(\text{red})} \] Substituting the values we have: \[ P(\text{ace} | \text{red}) = \frac{\frac{1}{26}}{\frac{1}{2}} = \frac{1}{26} \times \frac{2}{1} = \frac{2}{26} = \frac{1}{13} \]
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Convert the probability to a percentage: \[ \frac{1}{13} \approx 0.07692 \] To convert to a percentage, multiply by 100: \[ 0.07692 \times 100 \approx 7.692 % \]
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Round to the nearest tenth: Rounding 7.692% to the nearest tenth gives us: \[ 7.7% \]
Therefore, \( P(\text{ace} | \text{red}) \approx 7.7% \).