Problem 10 · 2017 AMC 8
Medium
Counting & Probability
careful-counting
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
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Answer: C — 3/10.
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Hint 1 of 2
"4 is the largest" is a quiet double condition: 4 must be in the draw, and 5 must be out. The other two cards are then forced to come from {1, 2, 3}.
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Hint 2 of 2
Translate 'largest is X' into 'include X, exclude everything bigger.' Then just count how to fill the remaining slots from what's left below X.
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Approach: include the max, fill the rest from below it
- Decode the condition: for 4 to be the biggest card drawn, you must take the 4 and avoid the 5. So the other two cards are chosen from {1, 2, 3}. That reframing is the insight.
- Favorable draws: choose 2 of {1, 2, 3} = C(3, 2) = 3 ways (namely {1,2,4}, {1,3,4}, {2,3,4}).
- All draws: C(5, 3) = 10. Probability = 3/10 = 3/10.
- Why this transfers: 'the max equals X' problems always split into 'X is in, everything above X is out' — the rest is a small count among the cards below X.
Another way — list the favorable subsets:
- With only 5 cards, just write the 3-card sets whose max is 4: {1,2,4}, {1,3,4}, {2,3,4} — exactly 3 of them.
- Out of C(5,3) = 10 equally likely draws, that's 3/10.
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