Problem 7 · 2009 Math Kangaroo
Medium
Counting & Probability
caseworkcomplementary-counting
In a drawer there are 2 white, 3 red and 4 blue socks. Lisa knows that one third of the socks have holes, but she does not know the colour of the faulty socks. She randomly picks socks from the drawer until she has a usable pair, i.e. a pair without holes and of equal colour. What is the minimum number of socks she has to draw to be certain of getting a usable pair?
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Answer: D — 7
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Hint 1 of 2
Picture an adversary who hands you the worst possible socks before you can win.
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Hint 2 of 2
The worst run is all faulty socks plus one good sock of every colour—count those, then add one.
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Approach: worst-case (pigeonhole) counting
- Three socks have holes; even if you draw all three, they cannot help.
- You could then draw one good sock of each colour (white, red, blue) without a matching pair: 3 more.
- After 3 + 3 = 6 socks you might still have no usable pair, so the 7th guarantees one. Answer 7.
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