To find the probability of being dealt either an eight or a black card from a standard 52-card deck, we can use the principle of inclusion-exclusion.
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Count the total number of favorable outcomes:
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Number of eights in a deck: There are 4 eights (one for each suit: hearts, diamonds, clubs, and spades).
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Number of black cards in a deck: There are 26 black cards (13 clubs and 13 spades).
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Number of eights that are black cards: There are 2 black eights (the eight of clubs and the eight of spades).
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Use the inclusion-exclusion principle: \[ P(\text{eight or black}) = P(\text{eight}) + P(\text{black}) - P(\text{eight and black}) \]
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Calculate each probability:
- \( P(\text{eight}) = \frac{4}{52} \)
- \( P(\text{black}) = \frac{26}{52} \)
- \( P(\text{eight and black}) = \frac{2}{52} \)
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Combine the probabilities: \[ P(\text{eight or black}) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} \] \[ P(\text{eight or black}) = \frac{4 + 26 - 2}{52} = \frac{28}{52} \]
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Simplify the fraction: \[ P(\text{eight or black}) = \frac{28}{52} = \frac{7}{13} \]
Thus, the probability that you are dealt an eight or a black card is \(\frac{7}{13}\).