1991 AIME Problems/Problem 10
Problem
Two three-letter strings, and
, 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
when it should have been a
, or as a
when it should be an
. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let
be the three-letter string received when
is transmitted and let
be the three-letter string received when
is transmitted. Let
be the probability that
comes before
in alphabetical order. When
is written as a fraction in lowest terms, what is its numerator?
Solution
Let us make a chart of values in alphabetical order, where are the probabilities that each string comes from
and
multiplied by
, and
denotes the partial sums of
(in other words,
):
\[\begin{tabular}{|r||r|r|r|} \hline \text{String}&P_a&P_b&S_b\\ \hline aaa & 8 & 1 & 1 \\ aab & 4 & 2 & 3 \\ aba & 4 & 2 & 5 \\ abb & 2 & 4 & 9 \\ baa & 4 & 2 & 11 \\ bab & 2 & 4 & 15 \\ bba & 2 & 4 & 19 \\ bbb & 1 & 8 & 27 \\ \hline \end{tabular}\] (Error compiling LaTeX. Unknown error_msg)
The probability is , so the answer turns out to be
, and the solution is
.
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |