{"status": "success", "data": {"description_md": "Raashan, Sylvia, and Ted play the following game. Each starts with \\$$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives \\$$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have \\$$1$? (For example, Raashan and Ted may each decide to give \\$$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have \\$$0$, Sylvia will have \\$$2$, and Ted will have \\$$1$, and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their \\$$1$ to, and the holdings will be the same at the end of the second round.)\n\n$\\textbf{(A) } \\frac{1}{7} \\qquad\\textbf{(B) } \\frac{1}{4} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{1}{2} \\qquad\\textbf{(E) } \\frac{2}{3}$", "description_html": "<p>Raashan, Sylvia, and Ted play the following game. Each starts with $<span class=\"katex--inline\">1</span>. A bell rings every <span class=\"katex--inline\">15</span> seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $<span class=\"katex--inline\">1</span> to that player. What is the probability that after the bell has rung <span class=\"katex--inline\">2019</span> times, each player will have $<span class=\"katex--inline\">1</span>? (For example, Raashan and Ted may each decide to give $<span class=\"katex--inline\">1</span> to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $<span class=\"katex--inline\">0</span>, Sylvia will have $<span class=\"katex--inline\">2</span>, and Ted will have $<span class=\"katex--inline\">1</span>, and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $<span class=\"katex--inline\">1</span> to, and the holdings will be the same at the end of the second round.)</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) } \\frac{1}{7} \\qquad\\textbf{(B) } \\frac{1}{4} \\qquad\\textbf{(C) } \\frac{1}{3} \\qquad\\textbf{(D) } \\frac{1}{2} \\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 10B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc10B_p23", "prev": "/problem/19_amc10B_p21"}}