One-One Correspondences Adam the ant starts at $(0,0)$. Each minute, he flips a fair coin. If he flips heads, he moves $1$ unit up; if he flips tails, he moves $1$ unit right.
Betty the beetle starts at $(2,4)$. Each minute, she flips a fair coin. If she flips heads, she moves $1$ unit down; if she flips tails, she moves $1$ unit left.
If the two start at the same time, what is the probability that they meet while walking on the grid?
How do I go about solving this problem, and what's the answer?

Adam and Betty may meet only after three steps from the start, in the points $(2,1),(1,2)$ or $(0,3)$.
After three steps, Adam is in $(2,1)$ with probability $\frac{3}{8}$, in $(1,2)$ with probability $\frac{3}{8}$ and in $(0,3)$ with probability $\frac{1}{8}$.
After three steps, Betty is in $(2,1)$ with probability $\frac{1}{8}$, in $(1,2)$ with probability $\frac{3}{8}$ and in $(0,3)$ with probability $\frac{3}{8}$.
So the probability that Adam and Betty meet is:
$$ \frac{3}{8}\cdot\frac{1}{8}+\frac{3}{8}\cdot\frac{3}{8}+\frac{3}{8}\cdot\frac{1}{8}=\color{red}{\frac{15}{64}}.$$
