MathDB

Each of the integers from

1

n

is written on a separate card, and then the cards are combined into a deck and shuffled. Three players,

A,B,

and

C,

take turns in the order

A,B,C,A,\dots

choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered

1.

Show that for each of the three players, there are arbitrarily large values of

n

for which that player has the highest probability among the three players of winning the game.

Problem Statement