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13 friends are sitting around a table. They shuffle a half-deck of cards, consisting of only the black cards of a standard deck, and deal out two cards to each person.

There are then a series of turns. On each turn, everyone takes the lower ranked of their two cards and passes it to the person on their right (A=1, J=11, Q=12, K=13). If both of a person's cards are the same rank, then the game immediately ends.

What is the probability that they never stop?

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  • $\begingroup$ Isnt this a duplicate / has been answered twice lol. $\endgroup$ – Daedric Jan 25 '16 at 17:39
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The chance is 0%

Let us begin with this observation: with 13 pair of cards in play, that accounts for every card in the half-deck. There are no cards remaining out of play.

So let's construct a scenario in which the game would never stop. To make that happen, each player must have one high card (say, above a 7) and one low card (7 or below). Since each player has a high card and a low card, and passes the low card, nobody will wind up with a pair, since all the low cards are moving at the same time, just chasing each other around the table, and never matching the (above a 7) high cards that aren't moving.

Sounds good, right?

The problem is that this precise situation can't actually occur; there are only 12 cards in the half-deck above a 7 (pairs of 8s, 9s, 10s, jacks, queens, and kings), but 13 players. Which means that we're guaranteed to have at least one player whose highest card is a 7 or lower.

Contrariwise, there are only 12 cards in the half-deck that are below a 7 (pairs of aces, 2s, 3s, 4s, 5s, and 6s). This means that we're guaranteed to have at least one player whose lowest card is a 7 or higher.

So over time, in this game, it's the lowest cards that will be circulating around the table: the pairs of aces through sixes, plus one seven. And since we're guaranteed that at least one player's highest card will be a 7 or lower which will remain in their hand as other cards are passed around the table, that player will eventually form a match, ending the game.

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The probability is zero, as shown in this answer to the same question with a deck of 25 pairs. By analogy to this answer the match will happen within $12$ passes

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