  A two lane road approaches a stoplight. The left lane merges into the right
just past the intersection. Vehicles approach the intersection one at a time,
with some drivers always choosing the right lane, while others always choose
the shorter lane, giving preference to the right lane to break ties. An arrival
sequence of vehicles can be represented as a binary string, where the zeros
represent drivers always choosing the right lane, and the ones represent
drivers choosing the shorter lane. From each arrival sequence we construct a
merging path, which is a lattice path determined by the lane chosen by each
car. We give closed formulas for the number of merging paths reaching the point
$(n,m)$ with exactly $k$ zeros in the arrival sequence, and the expected length
of the right lane for all arrival sequences with exactly $k$ zeros. Proofs
involve an adaptation of Andre's Reflection Principle. Other interesting
connections also emerge, including to: Ballot numbers, the expected maximum
number of heads or tails appearing in a sequence of $n$ coin flips, the largest
domino snake that can be made using pieces up to $[n:n]$, and the longest trail
on the complete graph $K_n$ with loops.
