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)