imc

Question

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

Accepted Answer

WLOG, we assume Sonya jumps 0.5 units every time, since that is her expected value. If Sonya is within 0.5 blocks of an edge, she can jump off the board. Let us examine the region that is at most 0.5 blocks from exactly one edge. FIGURE If Sonya starts in this region, she has a 14 chance of landing outside (there's exactly one direction she can hop to get out). The total area of this region is 40.55=10. For this region, Sonya has a 14 chance, so we multiply 10 by 14 to get 2.5. If Sonya is in one of the corner squares, she can go two directions to get out, so she has a 24=12 chance to get out. The total area is 0.50.54=1, so this region yields 121=12. Adding the two, we get 3. This is out of 36 square units of area, so our answer is thus 112.