Problem 26 · 2017 Math Kangaroo
Stretch
Counting & Probability
casework
Anna has five boxes, as well as five black balls and five white balls. She is allowed to decide how she shares out the balls between the boxes, as long as she puts at least one ball into each box. Beate randomly chooses one box and takes one ball without looking. Beate wins if she draws a white ball; otherwise Anna wins. How should Anna distribute the balls to get the highest probability of winning?
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Answer: D — Anna puts all of the white balls into one box and then puts one black ball into each box.
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Hint 1 of 2
Anna wins when Beate draws black, so Anna wants to minimise the chance of a white draw.
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Hint 2 of 2
Beate picks a box uniformly first; compute the white-draw probability for each option and pick the smallest.
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Approach: compute Beate's white-draw probability for each distribution
- Beate first picks one of the 5 boxes with equal chance \(\frac{1}{5}\), then a ball from it; Anna wants the white-draw probability as small as possible.
- Option D buries all 5 white balls in one box that also gets a black ball (6 balls, \(\frac{5}{6}\) white) while the other four boxes hold only black, giving white chance \(\frac{1}{5}\cdot\frac{5}{6} = \frac{1}{6}\).
- Every other option leaves more boxes containing white balls, so its white chance exceeds \(\frac{1}{6}\) (e.g. option C gives \(\frac{1}{5}\)).
- The smallest white chance, hence Anna's best play, is option D.
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