It is known that a reduced matrix obtained by Gaussian random projections preserves the stability of an original matrix in a probabilistic manner. However, due to unknown constants appearing in a probability concentration inequality, the exact value of the probability when stability is preserved, or the condition for the dimension of the reduced matrix to preserve stability with high probability, is not known. In this study, we derive the stable probability of the reduced matrix obtained by Gaussian random projections by presenting a variant of the theorem used for the existing research without unknown constants.