{"status": "success", "data": {"description_md": "Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\\tfrac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?\n\n$\\textbf{(A)}\\; \\dfrac{1}{2} \\qquad\\textbf{(B)}\\; \\dfrac{3}{5} \\qquad\\textbf{(C)}\\; \\dfrac{2}{3} \\qquad\\textbf{(D)}\\; \\dfrac{3}{4} \\qquad\\textbf{(E)}\\; \\dfrac{4}{5}$\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>Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is  <span class=\"katex--inline\">\\tfrac{1}{2}</span> , independently of what has happened before. What is the probability that Larry wins the game?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\; \\dfrac{1}{2} \\qquad\\textbf{(B)}\\; \\dfrac{3}{5} \\qquad\\textbf{(C)}\\; \\dfrac{2}{3} \\qquad\\textbf{(D)}\\; \\dfrac{3}{4} \\qquad\\textbf{(E)}\\; \\dfrac{4}{5}</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": 2, "problem_name": "2015 AMC 12B Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/15_amc12B_p10", "prev": "/problem/15_amc12B_p08"}}