Probability of meeting in the grid. Aubrey leaves home at $(0,0)$ to walk to school at $(1,5)$ and travels one block (one unit) north or east every minute; she does not leave the rectangle $0 \leq x \leq1$, $0\leq y \leq5$. Xander leaves school at the same time and is headed back toward home else at $(0,0)$ and travels one block (one unit) south or west every minute; he stays within the same rectangle. At each corner where she has a choice, Aubrey flips a coin to determine whether she should go north or east; at each corner where he has choice, Xander flips a coin to determine whether he should go south or west. Compute the probability that they meet.

First, note that the distance between home and school is $6$ units. Since they start at the same time and are moving with the same speed ($1$ unit per minute) , they can only meet after three minutes. So, the possible meeting points are $(0,3)$ and $(1,2)$. Let's find the probability that the meet in the first one. For that to happen Aubrey must move north 3 times in a row, that is a $0.5^3$ probability, while Xander must NOT move south 3 times in a row, that is a $1 - 0.5^3$. The probability that they meet at $(0,3)$ is the product of those two independent probabilities. To complete the answer note that the problem is symmetric so the probability that they meet at $(1,2)$ is the same.
