Alice's Cards
Source: 2023 AIME I #6
February 8, 2023
AIMEAIME Iprobabilityexpected valuenumber theoryrelatively prime
Problem Statement
Alice knows that red cards and 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 , where and are relatively prime positive integers. Find .