In network coding, for the case that the network topology is unknown completely, random linear network coding has been proposed as an acceptable coding technique. In this paper, we define average failure probability of random linear network coding in order to characterize the performance of random network coding, and then analyze this failure probability for different known topological information of network. We obtain several upper bounds on the failure probabilities, and further show that, for some networks, these upper bounds are tight or asymptotically tight. Moreover, if the more topological information of the network is utilized, the better upper bounds are acquired.