{"status": "success", "data": {"description_md": "Alice, Bob, and Carol play a game in which each of them chooses a real number between $0$ and $1.$ The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between $0$ and $1,$ and Bob announces that he will choose his number uniformly at random from all the numbers between $\\tfrac{1}{2}$ and $\\tfrac{2}{3}.$ Armed with this information, what number should Carol choose to maximize her chance of winning?\n\n$\\textbf{(A) }\\frac{1}{2}\\qquad<br>\\textbf{(B) }\\frac{13}{24} \\qquad<br>\\textbf{(C) }\\frac{7}{12} \\qquad<br>\\textbf{(D) }\\frac{5}{8} \\qquad<br>\\textbf{(E) }\\frac{2}{3}\\qquad$\n___\nFull 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>Alice, Bob, and Carol play a game in which each of them chooses a real number between  <span class=\"katex--inline\">0</span>  and  <span class=\"katex--inline\">1.</span>  The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between  <span class=\"katex--inline\">0</span>  and  <span class=\"katex--inline\">1,</span>  and Bob announces that he will choose his number uniformly at random from all the numbers between  <span class=\"katex--inline\">\\tfrac{1}{2}</span>  and  <span class=\"katex--inline\">\\tfrac{2}{3}.</span>  Armed with this information, what number should Carol choose to maximize her chance of winning?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }\\frac{1}{2}\\qquad\\textbf{(B) }\\frac{13}{24} \\qquad\\textbf{(C) }\\frac{7}{12} \\qquad\\textbf{(D) }\\frac{5}{8} \\qquad\\textbf{(E) }\\frac{2}{3}\\qquad</span> </p>&#10;<hr><p>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": "2018 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc12A_p25", "prev": "/problem/18_amc12A_p23"}}