Within the class of models producing upper and lower probability systems, as discussed in Dempster (1967a), a simple and important subclass may be characterized by random intervals on the line. Detailed expressions are given here for the upper and lower probabilities of a general fixed closed interval determined by a general random closed interval. Such random closed intervals occur in the applications of the general class of models to statistical inference described in Dempster (1966, 1967b, 1968). The illustration given here concerns inference about binomial $p$ and stresses the flexibility allowed in the introduction of prior information.