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Extreme electricity prices occur with a higher frequency and a larger magnitude in recent years. Accurate forecasting of the occurrence of extreme prices is of great concern to market operators and participants. This paper aims to forecast the occurrence probability of day-ahead extremely low and high electricity prices and investigate the relative importance of different influencing variables. The data obtained from the Australian National Electricity Market (NEM) were employed, including historical prices (one day before and one week before), reserve capacity, load demand, variable renewable energy (VRE) proportion and interconnector flow. A Multivariate Logistic Regression (MLgR) model was proposed, which showed good forecasting capability in terms of model fitness and classification accuracy with different thresholds of extreme prices. In addition, the performance of the MLgR model was verified by comparing with two other models, i.e., Multi-Layer Perceptron (MLP) and Radical Basis Function (RBF) neural network. Relative importance analysis was performed to quantify of the contribution of the variables. The proposed method enriches the theories of electricity price forecast and advances the understanding of the dynamics of extreme prices. By applying the model in practice, it will contribute to promoting the management of operation and establishment of a robust energy market.