Alex the ant starts at $(0,0)$. Each minute, he flips a fair coin. If he flips heads, he moves one unit up; if he flips tails, he moves one unit right. Betty the beetle starts at $(2,4)$. Each minute, she flips a fair coin. If she flips heads, she moves one unit down; if she flips tails, she moves one unit left. If the two start at the same time, what is the probability that they meet while walking on the grid?
They must meet at one of those points: $(0,3),(1,2),(2,1)$. each ant has a probability of $1/2^3$ to go through a specific path to one of the points, since they are 3-step from the start point.
Separate into complementary events:
meeting at $(0,3)$: "A" has 1 path, "B" has 3, we get: $P(0,3)=(1/8)*(3/8)$
meeting at $(1,2)$: Both has 3 paths: $P(1,2)=(3/8)*(3/8)$
meeting at $(2,1)$: same as the first
Summing the cases: $P=3/64 + 9/64 + 3/64 = 15/64$