1991 AIME Problem 10

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?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.