  A rhombus tiling of a hexagon is said to be centered if it contains the
central lozenge. We compute the number of vertically symmetric rhombus tilings
of a hexagon with side lengths $a, b, a, a, b, a$ which are centered. When $a$
is odd and $b$ is even, this shows that the probability that a random
vertically symmetric rhombus tiling of a $a, b, a, a, b, a$ hexagon is centered
is exactly the same as the probability that a random rhombus tiling of a $a, b,
a, a, b, a$ hexagon is centered. This also leads to a factorization theorem for
the number of all rhombus tilings of a hexagon which are centered.
