Problem 34 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiply
Roz bought 12 cups of yogurt, but 4 of them are spoiled. She grabs 3 cups at random, one after another. Find the probability that all 3 she grabs are spoiled.
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Answer: 1/55
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Hint 1 of 3
Grabbing cups one after another (without putting any back) is an AND process with changing counts.
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Hint 2 of 3
Track how many spoiled cups and how many total cups are left at each grab.
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Hint 3 of 3
Multiply \((4/12) \times (3/11) \times (2/10)\).
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Approach: AND process without replacement
- Grab three cups in a row (none put back); the spoiled count and the total both drop each time.
- Multiply: \(\frac{4}{12} \times \frac{3}{11} \times \frac{2}{10} = \frac{24}{1320} = \frac{1}{55}\).
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