Problem:
Objects and move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object starts at and each of its steps is either right or up, both equally likely. Object starts at and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since there are twelve steps between and and can meet only after they have each moved six steps. The possible meeting places are , and . Let and denote the number of paths to from and , respectively. Since has to take steps to the right and has to take steps down, the number of ways in which and can meet at is
Since and can each take paths in six steps, the probability that they meet is
OR
Consider the walks that start at , end at , and consist of 12 steps, each one either up or to the right. There is a one-to-one correspondence between these walks and the set of -paths where and meet. In particular, given one of the walks from to , the path followed by consists of the the first six steps of the walk, and the path followed by is obtained by starting at and reversing the last six steps of the walk. There are paths that take 6 steps from and paths that take 6 steps from , so there are pairs of paths that and can take. The probability that they meet is
The problems on this page are the property of the MAA's American Mathematics Competitions