MathDB

Felix is playing a card-flipping game.

n

face-down cards are randomly colored, each with equal probability of being black or red. Felix starts at the

1

st card. When Felix is at the

k

th card, he guesses its color and then flips it over. For

k < n

, if he guesses correctly, he moves onto the

(k + 1)

-th card. If he guesses incorrectly, he gains

k

penalty points, the cards are replaced with newly randomized face-down cards, and he moves back to card

1

to continue guessing. If Felix guesses the

n

th card correctly, the game ends. What is the expected number of penalty points Felix earns by the end of the game?

Problem Statement