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2023 AMC 10B Problem 19

Problem 19 of 25HarderProbability & StatisticsProblem-Solving Techniques

Sonya the frog chooses a point uniformly at random lying within the square [0,6]×[0,6][0, 6] \times [0, 6] in the coordinate plane and hops to that point. She then chooses a distance uniformly at random from [0,1][0, 1] and a direction uniformly at random from {north,south,east,west}.\{\text{north}, \text{south}, \text{east}, \text{west}\}. All her choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

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Solution

The four directions behave the same by symmetry, so say she hops east. She lands outside exactly when her xx-coordinate plus the hop distance dd tops 6.6. Fix d.d. Her xx-coordinate is uniform on [0,6],[0, 6], so it beats 6−d6 - d with probability d6.\frac{d}{6}. Now average over dd uniform on [0,1]:[0, 1]: 16⋅12=112.\frac{1}{6} \cdot \frac{1}{2} = \frac{1}{12}. Thus, B is the correct answer.
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Tagged: geometric probability · symmetry

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