AbstractAssume that the vertices of a graph G are always operational, but the edges of G are operational independently with probability p ∈ [0, 1]. For fixed vertices s and t, the two‐terminal reliability of G is the probability that the operational subgraph contains an (s, t)‐path, while the all‐terminal reliability of G is the probability that the operational subgraph contains a spanning tree. Both reliabilities are polynomials in p, and have very similar behavior in many respects. However, unlike all‐terminal reliability polynomials, little is known about the roots of two‐terminal reliability polynomials. In a variety of ways, we shall show that the nature and location of the roots of two‐terminal reliability polynomials have significantly different properties than those held by roots of the all‐terminal reliability polynomials.