For the risk multi-criteria decision-making problems in which the criteria value of alternatives are in the form of extended grey numbers, a grey stochastic multi-criteria decision-making approach is proposed based on the Hausdorff distance of extended grey numbers. Firstly, the discrete extended grey random variable as well as its expected value and standard deviation are defined. Then, motivated by the idea of the Hausdorff distance, the Hausdorff distance of extended grey numbers is defined. The expected value of each alternative under every criterion is calculated, and then the expected valued decision matrix is obtained. After that, a TOPSIS-based grey stochastic multi-criteria decision-making approach is built according to the idea of traditional TOPSIS method. Finally, an example is given to verify the effectiveness and feasibility of the developed approach.