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Difference between revisions of "1991 AIME Problems/Problem 10"

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(solution, but too lazy to finish typing)
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== Problem ==
 
== Problem ==
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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?
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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 [[irreducible fraction|lowest terms]], what is its [[numerator]]?
  
 
== Solution ==
 
== Solution ==
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{{solution}}
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Let us make a chart of values, where <math>P_a,\ P_b</math> are the probabilities that each string comes from <math>aaa</math> and <math>bbb</math> multiplied by <math>27</math>, and <math>\displaystyle X_b</math> denoting the sum of all of the previous terms of <math>\displaystyle P_b</math>:
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{| class= "wikitable" align="center"
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| String || <math>\displaystyle P_a</math> || <math>\displaystyle P_b</math> || <math>\displaystyle X_b</math>
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|-
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| aaa || 8 || 1 || 1
 +
|-
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| aab || 4 || 2 || 3
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|-
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| aba || 4 || 2 || 5
 +
|-
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| abb || 2 || 4 || 9
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|-
 +
| baa || 4 || 2 || 11
 +
|-
 +
| bab || 2 || 4 || 15
 +
|-
 +
| bba || 2 || 4 || 19
 +
|-
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| bbb || 1 || 8 || 27
 +
|}
 +
 
 +
The probability is <math>P_a \cdot (27 - X_b)</math> for each of the strings over <math>27^2</math>, so the answer turns out to be <math>\frac{8\cdot 26 + 4 \cdot 24 \ldots 2 \cdot 8 + 1 \cdot 0}{27^2} = \frac{532}{729}</math>, and the solution is <math>532</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1991|num-b=9|num-a=11}}
 
{{AIME box|year=1991|num-b=9|num-a=11}}
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 +
[[Category:Intermediate Combinatorics Problems]]

Revision as of 19:31, 11 March 2007

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 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 $\displaystyle p$ be the probability that $S_a^{}$ comes before $S_b^{}$ in alphabetical order. When $\displaystyle p$ is written as a fraction in lowest terms, what is its numerator?

Solution

Let us make a chart of values, where $P_a,\ P_b$ are the probabilities that each string comes from $aaa$ and $bbb$ multiplied by $27$, and $\displaystyle X_b$ denoting the sum of all of the previous terms of $\displaystyle P_b$:

String $\displaystyle P_a$ $\displaystyle P_b$ $\displaystyle X_b$
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

The probability is $P_a \cdot (27 - X_b)$ for each of the strings over $27^2$, so the answer turns out to be $\frac{8\cdot 26 + 4 \cdot 24 \ldots 2 \cdot 8 + 1 \cdot 0}{27^2} = \frac{532}{729}$, and the solution is $532$.

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
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