Problem 17 · 2019 Math Kangaroo
Hard
Counting & Probability
careful-counting
There are five balls in a box: four contain chocolate, and one contains a boiled sweet. Johann and Maria take turns drawing a ball from the box without replacing it. Whoever draws the boiled sweet wins. Johann starts. What is the probability that Maria wins?
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Answer: A — \(\dfrac{2}{5}\)
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Hint 1 of 2
The single boiled sweet is equally likely to be the 1st, 2nd, 3rd, 4th or 5th ball drawn.
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Hint 2 of 2
Maria draws on turns 2 and 4, so count her winning positions out of five.
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Approach: the sweet is equally likely in each draw position
- By symmetry the boiled sweet is equally likely to be the 1st, 2nd, 3rd, 4th or 5th ball drawn, each with probability \(\frac{1}{5}\).
- Johann draws on turns 1, 3 and 5; Maria draws on turns 2 and 4.
- Maria wins in exactly 2 of the 5 positions, so her probability is \(\frac{2}{5}\).
- Answer (A).
Mark:
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