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The formula for the exact union bound (EUB) on BER based on the exact closed-form pair-wise error probability (PEP) is most accurate for calculating the bit-error-rates (BERs) of space-time (S-T) codes in multiple-input multiple output (MIMO) systems. In this paper, it is shown that using the exact closed-form PEP formula for numerical calculations of the PEP or the EUB on BER could produce negative results. The formula for the exact closed-form PEP is therefore re-examined and explanations for the problem are given. A simple trick is suggested to reduce the effect of the problem. A new upper bound on PEP derived using the holder inequality is proposed to replace the exact closed-form PEP formula. Numerical calculation results show that the union bound based on the proposed new upper bound on PEP is a very tight bound to the EUB and doesn't have the problem of negative results.