The well-known Braess' paradox illustrates situations when adding a new link to a traffic network might increase congestion in the network. In this article, we announce a number of new results devoted to the probability of Braess' paradox to occur in the classical network configuration, with particular emphasis on the Erlang distribution of parameters of the travel time function. This distribution is important in the context of traffic networks. However, other distributions will be analysed as well because Braess' paradox can be observed in various applied contexts such as telecommunication networks and power transmission networks. Our results revealed that typical probabilities for Braess' paradox to occur in the classical network configuration do not exceed 10%, and they are very low for some distributions of the parameters of travel time functions. If the classical network configuration consists of motorway sections and class A roads and the parameters of the travel time functions are modelled by the Erlang-2 distribution, then the probability of Braess' paradox to occur is 6%.