2023 AMC 10B Problem 19

Sonya the frog chooses a point uniformly at random lying within the square

[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]

and a direction uniformly at random from

\{\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

x

-coordinate plus the hop distance

d

tops

6.

Fix

d.

Her

x

-coordinate is uniform on

[0, 6],

so it beats

6 - d

with probability

\frac{d}{6}.

Now average over

d

uniform on

[0, 1]:

\frac{1}{6} \cdot \frac{1}{2} = \frac{1}{12}.

Thus, B is the correct answer.