The objective of this paper is two-fold. Initially, we present an analytic technique to rapidly evaluate an approximation to the union bound on the bit error probability of turbo codes. This technique exploits the most significant terms of the union bound, which can be calculated straightforwardly by considering the properties of the constituent convolutional encoders. Subsequently, we use the bound approximation to demonstrate that specific punctured rate-1/2 turbo codes can achieve a lower error floor than that of their rate-1/3 parent codes. In particular, we propose pseudo-random puncturing as a means of improving the bandwidth efficiency of a turbo code and simultaneously lowering its error floor.