{"status": "success", "data": {"description_md": "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?\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Two three-letter strings, <span class=\"katex--inline\">aaa</span> and <span class=\"katex--inline\">bbb</span>, 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 <span class=\"katex--inline\">a</span> when it should have been a <span class=\"katex--inline\">b</span>, or as a <span class=\"katex--inline\">b</span> when it should be an <span class=\"katex--inline\">a</span>. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let <span class=\"katex--inline\">S_a</span> be the three-letter string received when <span class=\"katex--inline\">aaa</span> is transmitted and let <span class=\"katex--inline\">S_b</span> be the three-letter string received when <span class=\"katex--inline\">bbb</span> is transmitted. Let <span class=\"katex--inline\">p</span> be the probability that <span class=\"katex--inline\">S_a</span> comes before <span class=\"katex--inline\">S_b</span> in alphabetical order. When <span class=\"katex--inline\">p</span> is written as a fraction in lowest terms, what is its numerator?</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "1991 AIME Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/91_aime_p11", "prev": "/problem/91_aime_p09"}}