The Basics of Poker

No post last week. The author was a bit worried that someone had give Martin another box of silver, but in fact it was the something quite different. Enough personal stuff, onto the math.

Say one was to go play a game of Texas Hold'em. What should the player know? Blackjack and craps have fairly simple strategies to follow to maximize expected play, or rather to minimize expected losses. For blackjack all someone has to do is remember a simple table. Poker is different. The basic requirement is to know the rules. What hands beat which hands, the order in which people bet, and generally how the game unfolds. Once the basics are learned, the two most important concepts are the Fundamental Theorem of Poker and bluffing. The fundamental theorem dictates that a player play his hand as if he could see everyone's cards and always applied the correct pot odds. Every time a player does this he increases his expected gain. Every time a player fails to do this, he is losing money.

Bluffing adds depth to poker. A player cannot follow the fundamental theorem perfectly. He can only guess what his opponents have. If a player plays his hand based on the fundamental theorem and never bluffs, his opponents will be able to accurately guess what cards he has and put him at a disadvantage. The skill in poker comes from balancing the concepts of the fundamental theorem and bluffing.

Those are the basics. Next it is important to get a sense for how often certain hands occur. Playing repeated hands of poker will give a player a good feel. A player gets two cards before he has to decide whether to play the hand or fold. What types of hands should a player ante up for and how often do those hands occur? Let us say a player likes to play the following types of hands: a pair, cards of the same suit that are adjacent (suited connectors), and two high cards (two cards that come from the set of 10, Jack, Queen, King, and Ace). Those are generally regarded as good hands to ante up on as they can lead to strong hands. Let the sets be defined as A for pairs, B for suited connectors, and C for high cards. The probability of getting one of those hands is the union of those three sets:

P(A U B U C) = P(A) + P(B) + P(C) -[P(AB) + P(AC) + P(BC)] + P(ABC)

Where 'U' indicates a union of sets and 'AB, ABC etc.' indicates an intersection of sets. For unions of sets, the basic rule is to add the sets of odd intersections and subtract the even intersections.

P(A) = probability of getting a pair. The first card can be anything. For a pair to occur the second card had to be one of the 3 remaining cards from the deck of 51. The first card in P(B) can be any card. For any card, there are two remaining cards that are suited connectors. If the first card was the Ace of Spades, the second card has to be the King or 2 of Spades. P(C) can only have a 10, Jack, Queen, King or Ace for its first card and second card. There are 20 such cards in the deck and 19 remaining after the first one has been dealt.
P(A) = (52/52)*(3/51)
P(B) = (52/52)*(2/51)
P(C) = (20/52)*(19/51)

The intersection P(ABC) and P(AB) cannot occur since two cards cannot be both a pair and suited connectors. P(AC) is the set of cards that are pairs of 10, Jack, Queen, King, or Ace. There are 13 different types of cards and P(AC) makes up 5 of those types (10, 10; Jack, Jack etc). P(AC) can be expressed as the portion of A that falls into C. Likewise with P(BC) with the caveat that only 4/13 types of suited connectors of B occur in C (since 10, 9 and Ace, 2 do not occur in C).
P(AC) = P(A)*(5/13)
P(BC) = P(B)*(4/13)

Plugging those into the formula:
P(A U B U C) = 20.6%
Thus a player that plays these types of hands will play roughly one of every five hands. To keep opponents from getting a clear read on what type of hands you prefer it may be advisable to play a junk hand occasionally.

I like playing suited connectors and pairs because if the next five cards improve your hand, a clear advantage can emerge. For example, if you play a suited connector and three of the next five cards are the same suit you get a flush. Unless there is a pair among those five cards, a flush will very likely be the best hand. But given that you played a pair or suited connector, what are the odds that the next five cards will improve your hand? I will attempt this question in the next post.