In this paper, making use of recent statistical physics techniques and models, we address the specific role of randomness in financial markets, both at the micro and themacro level. In particular, we review some recent results obtained about the effectiveness of random strategies of investment, compared with some of the most used trading strategies for forecasting the behaviour of real financial indexes.We also push forward our analysis by means of a self-organised criticality model, able to simulate financial avalanches in trading communitieswith different network topologies,where a Pareto-like power law behaviour of wealth spontaneously emerges. In this context, we present new findings and suggestions for policies based on the effects that random strategies can have in terms of reduction of dangerous financial extreme events, i.e. bubbles and crashes.