A pool design is random if it varies according to a probability distribution. There are four types of random design proposed in the literature: random incidence design, random k-set design, random distinct k-set design, and random k-size design. Recently Hwang gave an approximation to estimate the number of unresolved positives for random distinct k-set design. In this article, we give exact formulas for all four types of random designs for estimating the number of unresolved positives. We also do some numerical comparisons of the four designs.