Problem 27 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplylogical-reasoning
An urn has 3 red marbles and 1 blue marble. Two marbles are drawn at random. Find the probability that both are red if (a) the first marble is put back before the second draw; (b) the first marble is NOT put back.
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Answer: (a) 9/16; (b) 1/2
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Hint 1 of 4
Both draws happen, so this is an AND process. The key question: does the first draw change what's left for the second?
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Hint 2 of 4
With the marble put back, the second draw is just like the first (3 red out of 4). Multiply the two single-draw chances.
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Hint 3 of 4
Without putting it back, after pulling a red there are only 2 reds left out of 3 marbles. Multiply the changed chances.
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Approach: AND process for probabilities (with and without replacement)
- (a) With replacement (independent draws): each draw is red with probability \(\frac{3}{4}\), so \(\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}\).
- (b) Without replacement: the first is red with probability \(\frac{3}{4}\); after removing one red, 2 reds remain among 3 marbles, so the second is red with probability \(\frac{2}{3}\).
- Multiply: \(\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}\).
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