MathDB
Alice's Cards

Source: 2023 AIME I #6

February 8, 2023
AIMEAIME Iprobabilityexpected valuenumber theoryrelatively prime

Problem Statement

Alice knows that 33 red cards and 33 black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.