Answer:
Poker is one of the many games involving the use of a 52-card deck of playing
cards. The 52 cards are categorized by 13 ranks from Two through Ace (Aces can be
counted as both higher than King and lower than Two when needed, but can only count
as one at a time in a hand), and by four suits: diamonds, hearts, spades, and clubs. In the
game of poker, players attempt to assemble the best five-card hand according to the
definitions of each hand that can be made.
There are ten hands that can be made:
1) Royal Flush – all five cards are of the same suit and are of the sequence
10 – J – Q – K – A
2) Straight Flush – all five cards are of the same suit and are sequential in rank
(note that a royal flush is simply the highest-ranked straight flush)
3) Four-of-a-Kind (which will be abbreviated in this paper as 4OAK) – a hand
where four cards are all of the same rank
4) Full House – a hand consisting of one pair and a three-of-a-kind of a different
rank than the pair
5) Flush – all five cards are of the same suit but not all sequential in rank
6) Straight – all five cards are sequential in rank but are not all of the same suit
7) Three-of-a-Kind (which will be abbreviated as 3OAK) – a hand where three
cards are all of the same rank and the other two are each of different ranks from the
3OAK and each other
8) Two Pair – two pairs of two cards of the same rank (the ranks of each pair are
different in rank, obviously, to avoid a 4OAK)
9) One Pair – only two cards of the five are of the same rank with the other three
cards all having different ranks from each other and from that of the pair
10) High Card – a hand in which no better hand was made (i.e., one in which each
card is of a different rank than any other card and not all five are of the same suit or
sequential in rank
Poker games have many variations, some of which will be investigated here. One
such variation is “stud” poker in which a player must hold all the cards he/she is given.
This is opposed to “draw” poker in which a player can draw any number of replacement
cards after being dealt an initial five in the attempt to improve his/her hand. Texas Hold
em is another variation in which each player is only dealt two cards to themselves, but
through the course of the betting rounds a total of five cards are dealt as “community”
cards that any player can use with any combination of their two to make the best five-
card hand possible. Other variations include the use of jokers and wild cards. In this
paper I will derive the probabilities of being dealt one of the given hands in five-card stud
poker and how those probabilities change when jokers and wild cards are included. I will
also analyze Texas Hold em and derive the probability of a given hand winning
throughout the course of a few example games.
