Two three-letter strings, $$aaa^{}_{}$$ and $$bbb^{}_{}$$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $$a^{}_{}$$ when it should have been a $$b^{}_{}$$, or as a $$b^{}_{}$$ when it should be an $$a^{}_{}$$. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $$S_a^{}$$ be the three-letter string received when $$aaa^{}_{}$$ is transmitted and let $$S_b^{}$$ be the three-letter string received when $$bbb^{}_{}$$ is transmitted. Let $$p$$ be the probability that $$S_a^{}$$ comes before $$S_b^{}$$ in alphabetical order. When $$p$$ is written as a fraction in lowest terms, what is its numerator?

(第九届AIME1991 第10题)