Probabilities in Poker
Poker is a card game with many different variants, some of which can be fairly
complicated. Here, however, we play 5-card stud poker, one of the simplest variant of
the game. In 5-card stud poker a player is dealt 5 cards from a standard deck of 52
playing and he/she must try to make a winning hand out of these cards (so no card is
replaced).
A standard deck consists of 4 suits: diamonds, hearts, spades and clubs. Each suit has
the same 13 “kinds” of cards which, ranked from lowest to highest, are given as follows:
two, three, four, five, six, seven, eight, nine, ten, jack, queen, king and ace.
A poker hand is any hand that has a special pattern, such as one having exactly 2 cards
from the same “kind” of cards (this hand is called a pair; it’s the lowest-ranked poker
hand), or one having all cards from the same suit (this hand is called a flush; it’s ranked
high), or one having 4 cards from the same “kind” of cards (this hand is called a four-ofa-kind; it’s ranked very high), and so on. For more information on the types of poker
hands that exist – along with their respective rankings – go to
http://www.handsofpoker.net/.
Playing 5-card stud is what a probability theorist would call a fair experiment, since all
the hands that can be dealt in this poker game (the outcomes of the experiment) are
equally likely to occur.
The sample space S in this experiment (the game) is then the set of all 5-card hands
that can be dealt.
The number of such hands is given by
.
Note that the enumeration here involves combinations - not permutations – because
the order in which the 5 cards are dealt to a player is not important.
Let us now consider the following 4 events, each corresponding to a poker hand:
is the event that the hand contains exactly one pair (i.e. 2 cards from the same “kind”
of cards and all other cards from distinct “kinds” of cards).
is the event that the hand contains exactly two pairs, where each pair and the
remaining card all belong to distinct “kinds” of cards.
is the event that the hand contains a three-of-a-kind (i.e. 3 cards from the same
“kind” of cards and the remaining 2 cards from distinct “kinds” of cards).
is the event that the hand contains a full house (i.e. a pair and a three-of-a-kind).
We will now compute the probabilities associated with these 4 events using the formula
,
where
.
Here n(Ei) and n(S) denote, respectively, the number of hands in event E i (where
) and the number of hands in the sample space S.
To determine these probabilities correctly, we will examine the pattern of the poker hand
for each event and then carefully apply the counting principle using appropriate
combinations at each stage of the experiment. Remember here to think about the
possible ways in which all the cards in the hand can be picked.
One Pair (E1)
This is the hand with the pattern AABCD, where A, B, C and D are all from distinct
“kinds” of cards.
Applying the fundamental counting principle, we then need to pick a pair from one of the
13 “kinds” of cards (
) and then pick three cards, each belonging to one of the
remaining 12 “kinds” of cards
.
This yields
.
Therefore
.
So a player has roughly a 42% chance of drawing exactly one pair in 5-card stud
poker. This is the lowest possible poker hand that he/she can draw (not counting single
high cards).
Two Pairs (E2)
This is the hand with the pattern AABBC, where A, B and C are all from distinct “kinds” of
cards.
Applying the fundamental counting principle, we then need to pick two pairs from two of
the 13 “kinds” of cards
and then pick a card belonging to one of the
remaining 11 “kinds” of cards
.
This yields
.
Therefore
.
So a player has roughly a 5% chance (or 1 in 20) of drawing exactly two pairs in 5card stud poker. This is the second lowest possible poker hand that he/she can draw
(not counting single high cards).
Three-of-a-Kind
This is the hand with the pattern AAABC, where A, B and C are all from distinct “kinds” of
cards.
Applying the fundamental counting principle, we then need to pick a three-of-a-kind from
one of the 13 “kinds” of cards (
) and then pick two cards, each belonging to
one of the remaining 12 “kinds” of cards
.
This yields
.
Therefore
.
So a player has roughly a 2% chance (1 in 50) of drawing a three-of-a-kind in 5-card
stud poker.
Full House
This is the hand with the pattern AABBB, where A and B are all from distinct “kinds” of
cards.
Applying the fundamental counting principle, we then need to pick a pair from one of the
13 “kinds” of cards
and then pick a three-of-a-kind from one of the
remaining 12 “kinds” of cards
.
This yields
.
Therefore
.
So a player has roughly a 0.1% chance (1 in 1,000) of drawing a full-house in 5-card
stud poker.