Problem of the Week 1102

The Holy Game of Poker*

Consider a 54-card deck consisting of the usual 52 cards and two jokers, which are wild, meaning that they can represent any of the 52 standard cards. For a standard deal of five cards from the 54, which poker hand is more likely: four of a kind or a full house (three of a kind and a pair)?

Note: "Four of a kind" refers to a true four of a kind, and excludes five of a kind. And each hand is assigned to its optimal possibility according to the rules of poker: so A-A-A-2-Joker must be taken to be four of a kind (not a full house); and 3-3-3-3-Joker would count has 5 of a kind, not 4 of a kind.

Aside: This reminds me of the famous puzzle: If one is playing poker with a 52-card deck (five-card stud, meaning five cards are dealt and there are no exchanges) and God offers you your choice of a full house, which full house should you choose? Hint: Choosing three aces and two kings would NOT be the best choice.

Source: The book Impossible by J Havil, Princeton University Press, 2008.

© Copyright 2008 Stan Wagon. Reproduced with permission.

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* Title from an old Leonard Cohen song, The Stranger.