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The dielectric breakdown strength of the composite with a boron nitride as an insulating filler and/or a carbon black as an electrically conductive filler was measured and discussed from the view of percolation phenomena. The dielectric breakdown strength decreases with increasing the volume of carbon black. It is dominated by finite clusters of a carbon black and is not influenced by a boron nitride. The dielectric breakdown strength is described by a function of a power law of the distance from the volume fraction of carbon black to the percolation threshold with the critical exponent of ν = 0.9. And it was found that the percolation threshold of the composite with a carbon black and a boron nitride was larger than that of the composite with a carbon black.