Probabilities in Poker

Here we are playing 5-card stud, one of the simplest version of poker games in which a player is dealt5 cards from a standard deck of 52 playing cards.

A standard deck consists of 4 suits (diamonds, hearts, spades and clubs), where each suit has the same 13 “kinds” of cards: aces, twos, threes, fours, fives, sixes, sevens, eights, nines, tens, jacks, queens, and kings.

Playing 5-card stud is what a probability theorist would call afair experiment, since all the hands that can be dealt in this poker game (i.e. the outcomes of the experiment) are equally likely to occur.

The sample space 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 is not important.

Let us now consider the following 4events:

  • is the event that the hand contains a single pair (i.e.2cards from the same “kind” of cards).
  • is the event that the hand contains twopairs, where each pair belongs 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).
  • 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 .

Single Pair

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 theremaining 12 “kinds” of cards .

This yields

.

Therefore

.

Two Pairs

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 oneof the remaining 11 “kinds” of cards .

This yields

.

Therefore

.

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

.

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

.