1991 AIME Problem 10

Two three-letter strings, a a a and b b b, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a \frac{1}{3} chance of being received incorrectly, as an a when it should have been a b, or as a b when it should have been 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 threeletter string received when aaa is transmitted and let S_{b} be the three-letter string received when b b b 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?