MathDB
Coin games

Source: 2017 AMC 10A #18

February 8, 2017
AMCAMC 10AMC 10 A2017 AMC 10A

Problem Statement

Amelia has a coin that lands heads with probability 13\frac{1}{3}, and Blaine has a coin that lands on heads with probability 25\frac{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 pq\frac{p}{q}, where pp and qq are relatively prime positive integers. What is qpq-p?
<spanclass=latexbold>(A)</span>1<spanclass=latexbold>(B)</span>2<spanclass=latexbold>(C)</span>3<spanclass=latexbold>(D)</span>4<spanclass=latexbold>(E)</span>5<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>3\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>5