Bluetooth is a promising wireless technology enabling (portable) devices to form short-range star-shaped wireless networks (or wireless personal area networks - PAN). Bluetooth relies on a frequency hopping physical layer, implying that hosts are not able to communicate unless they have previously discovered each by synchronizing their frequency hopping patterns. This further implies, that even if all nodes are within direct communication range, only those nodes that are synchronized within the personal area network can over-hear a transmission. To support any-to-any communication such personal area networks (or in Bluetooth terminology piconets and possibly scatternets) need to be established among nodes. Thus, it is of utmost importance to be able to model or predict the time required for nodes to discover each other, i.e., to synchronize them. This paper provides a mathematical analysis and corresponding simulation results for the Bluetooth discovery (inquiry) time in a fully connected (personal area network) situation with the population as a parameter. The simplification to the analysis and simulation is due to the intermediate train changes in the inquiry process as it is outlined in the paper.