{"status": "success", "data": {"description_md": "Real numbers between $0$ and $1$, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $0$ if the second flip is heads and $1$ if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \\tfrac{1}{2}$?\n\n$\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}$\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>Real numbers between  <span class=\"katex--inline\">0</span>  and  <span class=\"katex--inline\">1</span> , inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is  <span class=\"katex--inline\">0</span>  if the second flip is heads and  <span class=\"katex--inline\">1</span>  if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval  <span class=\"katex--inline\">[0,1]</span> . Two random numbers  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span>  are chosen independently in this manner. What is the probability that  <span class=\"katex--inline\">|x-y| &gt; \\tfrac{1}{2}</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}</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": 3, "problem_name": "2019 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p21", "prev": "/problem/19_amc12A_p19"}}