We use Erman and Wood's semiample extension of Poonen's closed point sieve to compute the probability that a semiample complete intersection over a finite field is smooth. This generalizes work of Bucur and Kedlaya, who provided the analogous calculation in the ample setting. We further extend the result by allowing the requirement that the complete intersection meet a closed subscheme transversely, so long as the subscheme satisfies a mild Altman Kleiman type condition. In both cases the probability stabilizes to a product of local factors determined by the semiample divisor in question.