Difference between revisions of "1991 AIME Problems/Problem 10"
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== Problem == | == Problem == | ||
+ | Two three-letter strings, <math>aaa^{}_{}</math> and <math>bbb^{}_{}</math>, 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 <math>a^{}_{}</math> when it should have been a <math>b^{}_{}</math>, or as a <math>b^{}_{}</math> when it should be an <math>a^{}_{}</math>. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let <math>S_a^{}</math> be the three-letter string received when <math>aaa^{}_{}</math> is transmitted and let <math>S_b^{}</math> be the three-letter string received when <math>bbb^{}_{}</math> is transmitted. Let <math>\displaystyle p</math> be the probability that <math>S_a^{}</math> comes before <math>S_b^{}</math> in alphabetical order. When <math>\displaystyle p</math> is written as a fraction in lowest terms, what is its numerator? | ||
== Solution == | == Solution == | ||
+ | {{solution}} | ||
== See also == | == See also == | ||
− | + | {{AIME box|year=1991|num-b=9|num-a=11}} |
Revision as of 02:28, 2 March 2007
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
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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 |