{"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$\\textbf{(A) } \\frac{1}{3} \\qquad \\textbf{(B) } \\frac{7}{16} \\qquad \\textbf{(C) } \\frac{1}{2} \\qquad \\textbf{(D) } \\frac{9}{16} \\qquad \\textbf{(E) } \\frac{2}{3}$", "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;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2019 AMC 10A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc10A_p23", "prev": "/problem/19_amc10A_p21"}}