I am trying to understand the Chinese Restaurant process (CRP) and Weighted Chinese Restaurant process (WCRP) described in a research paper "Automatic Discovery of Cognitive Skills"- Robert V. Lindsey, Mohammad Khajah, Michael C. Mozer to Improve the Prediction of Student Learning. In CRP all the implementations (cf., Infinite mixture models with nonparametric Bayes and the Dirichlet process) have a comparison made with the random number to decide if the customer chooses to sit on a new or existing table. Why this check is made and also how will this check condition differ in WCRP?
This implementation is using the Polya urn representation of the Dirichlet process like described by Blackwell and MacQueen (1973). In the link you've provided this particular part of the process is described as "With probability α/(1+α) he sits down at a new table." Conceptually one can think of this as capturing the idea that in principle there are an infinite number of possible tables to join.
The only difference under a weighted Chinese restaurant process in terms of the random number check is the probability of deciding to start a new table (cluster) will be different.