In this paper, we provide a new approach to the analytical estimation of the bit-error rate (BER) for convolutional codes for Viterbi decoding in the binary symmetric channel (BSC). The expressions we obtained for lower and upper BER bounds are based on the active distances of the code and their distance spectrum. The estimates are derived for convolutional codes with the rate $R=\frac{1}{2}$ but can be easily generalized for any convolutional code with rate $R=\frac 1n$ and systematic encoder. The suggested approach is not computationally expensive for any crossover probability of BSC channel and convolutional code memory, and it allows to obtain precise estimates of BER.