Problem 28 · AMC 8 Stretch
Stretch
Counting & Probability
or-process-addlogical-reasoning
Maile pulls one card at random from a standard 52-card deck. (a) What is the probability it is a seven or an ace? (b) What is the probability it is a seven or a red card?
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Answer: (a) 2/13; (b) 7/13
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Hint 1 of 3
First ask: can the two things happen at the same time on one card?
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Hint 2 of 3
A card can't be both a seven and an ace, so just add those chances. But a card CAN be both a seven and red (like the seven of hearts).
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Hint 3 of 3
(a) Add \(4/52 + 4/52\). (b) Add the sevens and the reds, then subtract the red sevens you counted twice.
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Approach: OR process — add, subtracting any overlap
- (a) Seven or ace: no card is both, so the chances simply add. There are 4 sevens and 4 aces: \(\frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}\).
- (b) Seven or red: there are 4 sevens and 26 red cards, but the seven of hearts and seven of diamonds got counted in BOTH groups. Subtract those 2 once: \(\frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}\).
- When two events can't overlap, just add; when they can overlap, subtract the part counted twice.
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