A method was previously developed for computation of pattern probabilities in random sequences under Markov chain models. We extend this method to the calculation of the joint distribution for two patterns. An application yields the distribution of the right choice measure for expressivity and how significance bounds depend on sequence length. These bounds are used to show that the choice of pyrimidine in codon position 3 of Escherichia coli genes deviates considerably from a general Markov process model for coding regions. We also derive some statistical evidence that this significant deviation is limited to codon position 3.