  We state, discuss, provide evidence for, and prove in special cases the
conjecture that the probability that a random tiling by rhombi of a hexagon
with side lengths $2n+a,2n+b,2n+c,2n+a,2n+b,2n+c$ contains the (horizontal)
rhombus with coordinates $(2n+x,2n+y)$ is equal to ${1/3} + g_{a,b,c,x,y}(n)
{\binom {2n}{n}}^3 / \binom {6n}{3n}$, where $g_{a,b,c,x,y}(n)$ is a rational
function in $n$. Several specific instances of this "1/3-phenomenon" are made
explicit.
