The use of a filler material in epoxy composite materials is an essential condition for reducing the unit cost of production and reinforcing mechanical strength. However, the dielectric strength of insulators decreases rapidly due to interactions between the epoxy resin and filler particles. In contrast to existing composite materials, nano-composite materials have superior dielectric strength, mechanical strength, and enduring chemical properties due to an increase in the bond strength of the polymer and nano material, It is reported that nano-fillers provide new characteristics different from the properties of the polymer material. This study is to improve the insulation capability of epoxy resins used in the insulation of a power transformer apparatus and many electronic devices mold. To accomplish this, the additional amount of nano-$SiO_2$ to epoxy resin was changed and the epoxy/$SiO_2$ nano composite materials were made, and the fundamental electrical properties were investigated using a physical properties and an analysis breakdown test. Using allowable breakdown probability, the optimum breakdown strength for designing an electrical apparatus was determined. The results found that the electrical characteristics of the nano-$SiO_2$ content specimens were superior to the virgin specimens. The 0.4 wt% specimens showed the highest electrical properties among the specimens examined with an allowable breakdown probability of 20 %, which indicates stable breakdown strength in insulating machinery design.