Problem 8 · 2018 Math Kangaroo
Medium
Counting & Probability
careful-counting
There are 65 balls in a box, 8 of which are white and the rest black. Up to 5 balls can be taken out of the box in one draw, and no balls may be put back. What is the minimum number of draws needed to be certain that at least one white ball is drawn?
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Answer: B — 12
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Hint 1 of 2
Imagine the unluckiest possible sequence of draws that keeps avoiding white balls.
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Hint 2 of 2
How many draws of up to 5 can be all black before white is forced?
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Approach: worst-case (pigeonhole) reasoning on the 57 black balls
- There are 57 black balls; in the worst case you keep pulling black.
- Eleven draws of 5 remove only 55 balls, so black balls might still remain; a twelfth draw can no longer be all black (at most 2 black left) and must include a white.
- Thus 12 draws guarantee at least one white ball.
- Answer: 12.
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