Abstract. 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 \frac{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.