In this paper we consider a time slotted cognitive radio (CR) network with N wireless channels and M secondary users (SUs). We focus on a multiple channel access policy where each SU stochastically decides whether to access idle channels or not based on the given access probability (AP) that is adapted to the channel state information (CSI), if possible. The AP plays an important role in the random access policy because it can control the number of SUs who can access idle channels in a simple manner and hence alleviate packet collisions among SUs. We assume that each SU can access at most L idle channels simultaneously at a time slot whenever possible. We consider three cases—(a) all SUs have full CSI, (b) all SUs have no CSI, and (c) all SUs have partial CSI. We analyze the throughput of an arbitrary SU for the three cases, and rigorously derive a closed-form expression of the optimal AP values that maximize the throughput of an arbitrary SU for the three cases. From the analysis, we show the impact of multiple channel access and the acquisition of CSI on throughput performance.