My first instinct was symmetry between events 1 and 2, which felt right but I fumbled explaining when exactly they'd differ based on starting positions.
First, clarify the assumptions: the walkers move independently, each step is to a uniformly random adjacent cell, and they stop when they exit the grid. Then, reason about the symmetry of the exit probabilities and the meeting probability, using the fact that the grid is finite and the walkers are independent. Finally, rank the events based on the relative magnitudes of these probabilities, justifying with intuitive arguments or simple calculations.
Pro tip: Emphasize that the exit probabilities depend only on the starting positions and the grid boundaries, not on the other walker, so the two exit events are mutually exclusive and their probabilities sum to 1. This immediately implies that one of them is at least 0.5, while the meeting probability is typically much smaller, so the ranking is clear.
State that the walkers move independently, each step to a uniformly random adjacent cell, and that 'exits the grid' means stepping outside the 10x10 boundaries. Define 'occupy the same cell at some point' as being on the same cell at the same time step while both are still inside.
Explain that the exit time for each walker depends only on its starting position and the grid boundaries. Since the walkers are independent, the events 'A exits before B' and 'B exits before A' are complementary (ignoring ties), so their probabilities sum to 1. Thus, one of them is at least 0.5.
Argue that the probability of the two walkers ever occupying the same cell while both are inside is generally small, especially on a 10x10 grid with random movement. Use the fact that they start at different positions and the grid is large relative to the number of steps before exit.
Compare the magnitudes: the larger of the two exit probabilities is at least 0.5, while the meeting probability is likely much less than 0.5. Therefore, the ranking is: the more likely exit event (depending on starting positions) first, the other exit event second, and the meeting event last.
If the starting positions are symmetric, the two exit probabilities are equal (each 0.5). Otherwise, the walker closer to the boundary is more likely to exit first. In either case, the meeting probability is smaller than the larger exit probability, so the ranking holds.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.