Abstract: Let S be a mixing shift of finite type and let W_n(S) be its set of words of length n. Define a random subset E of W_n(S) by choosing words from W_n(S) (independently, with some probability alpha). Let T be the SFT built from the set E. As n tends to infinity, what can be said about the likely properties of T as a function of alpha? We can give answers with regard to the emptiness and entropy of T. Also, for near 1, the likelihood that T has a unique irreducible component of positive entropy converges exponentially to 1 as n tends to infinity. Note that this version of "random SFT" differs from a notion by the same name studied by Bogenschutz and Gundlach, Kifer, and others, in the context of random dynamical systems or "bundled dynamical systems."