Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let be countable V‐structure which is homogeneous, simple and 1‐based. The first main result says that if is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if is “coordinatized” by a set with SU‐rank 1 and there is no definable (without parameters) nontrivial equivalence relation on (...) M with only finite classes, then is strongly interpretable in a random structure. (shrink)