{"status": "success", "data": {"description_md": "Amelia has a coin that lands heads with probability $\\tfrac{1}{3}$, and Blaine has a coin that lands on heads with probability $\\tfrac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $q-p$?\n\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5$", "description_html": "<p>Amelia has a coin that lands heads with probability  <span class=\"katex--inline\">\\tfrac{1}{3}</span> , and Blaine has a coin that lands on heads with probability  <span class=\"katex--inline\">\\tfrac{2}{5}</span> . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is  <span class=\"katex--inline\">\\tfrac{p}{q}</span> , where  <span class=\"katex--inline\">p</span>  and  <span class=\"katex--inline\">q</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">q-p</span> ?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ 5</span> </p>\n<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": "2017 AMC 10A Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/17_amc10A_p19", "prev": "/problem/17_amc10A_p17"}}