Problem:
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 31 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 Sa be the threeletter string received when aaa is transmitted and let Sb be the three-letter string received when bbb is transmitted. Let p be the probability that Sa comes before Sb in alphabetical order. When p is written as a fraction in lowest terms, what is its numerator?
Solution:
Let Sa, the three-letter string received when aaa is transmitted, be x1x2x3 and let Sb be y1y2y3, where each of the xk,yk is an a or a b. It will be convenient to introduce the symbol ≺ to denote that one string of letters precedes another in alphabetical order. (Thus, if S1 and S2 are two strings of letters, then S1≺S2 is to be read "S1 precedes S2 alphabetically.") We will find the probability that Sa≺Sb. Since the reception of any one letter is independent of that of any of the other letters, we have
Prob(Sa≺Sb)=Prob(x1x2x3≺y1y2y3)=Prob(x1≺y1)+Prob(x1=y1 and x2≺y2)+Prob(x1=y1 and x2=y2 and x3≺y3)=Prob(x1≺y1)+Prob(x1=y1)⋅Prob(x2≺y2)+Prob(x1=y1)⋅Prob(x2=y2)⋅Prob(x3≺y3).(∗)
Now x1≺y1 is true if and only if x1=a and y1=b; that is, if and only if these leading letters were received correctly. Since for each letter there is a 32 probability that it was received correctly, we conclude that
Prob(x1≺y1)=32⋅32=94.
Similarly, Prob(x2≺y2)=Prob(x3≺y3)=94. The relation x1=y1 is true if and only if one of these letters was received correctly and the other was received incorrectly. Thus
Prob(x1=y1)=Prob(x1=y1=a)+Prob(x1=y1=b)=32⋅31+31⋅32=94.
Identical reasoning shows that Prob(x2=y2)=94 also. Substituting these probabilities in (∗) we have
Prob(Sa≺Sb)=94+(94)2+(94)3=729532
The desired numerator is 532.
The problems on this page are the property of the MAA's American Mathematics Competitions