Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold'em is the version played most often at casinos and is the most popular among the "community cards" variants of poker. It is also the variant played at the World Series of Poker and on the World Poker Tour.
Each hand is played with a whole deck. One player is the dealer (this role rotates clockwise after each hand), and bets are placed in a clockwise order starting with the player on the dealer's left. Each hand has four stages, and after each stage there is a round of betting. The four stages are:
The object of the game is to form the best five-card hand possible using the player's two cards and the five "community cards" dealt in the middle of the table. A hand is won by having the best hand among the players who did not fold (i.e. refuse to match an opponent's bet, as described below), or by having everyone else fold.
We are going to use a $1/$2 betting structure. Before the pre-flop, the two players to the left of the dealer must bet $1 (these mandatory bets are called blinds, since the player must make them before she sees her cards). Then, following the betting order, each player may raise the bet, up to four times per player per betting round. (The blinds act as a bet, so in the pre-flop betting round, the first player to act will be the person three seats to the left of the dealer). Whenever a player raises the bet, the other players must call (that is, accept the raise), fold (that is, give up and lose the money already bet) or raise the bet even more. On the pre-flop and flop, the players bet $1 at a time, while on the turn and river they bet $2 at a time.
The hand ends when all but one player has folded or when all the cards have been dealt and the last betting round is over. In this last case, the players must show their cards and the player with the highest hand wins.
From highest to lowest, the possible five card hands in poker are ranked as follows:
There is no ranking of suits in poker, so two players who have identical hands but in different suits tie the hand and split the pot. The two cards that a player does not use in making his five card hand are ignored; they are not used to break ties between five card hands.
Pot odds are the odds you get when you analyze the current size of the pot against the cost of your next call. The general idea is to compare your chance of winning to your pot odds. You have good pot odds if your chance of winning is significantly bigger than the ratio of the bet to the pot size. For example, say you are on the turn, you have two hearts in your hand, and you have one opponent still in the hand. The community cards have two hearts, so any of the nine remaining hearts finishes a flush for you. We say that you have 9 "outs" (outs are the cards still unseen that will improve your hand) out of a total of 46 unseen cards. The ratio 9/46 is approximately 1/5. Suppose your opponent raises $2 and the pot you get if you call and win is $20. The ratio 2/20 is 1 in 10, which is smaller than your 1 in 5 chance of hitting the flush, so pot odds say that calling is the right move.
Implied odds take into account the fact that betting will continue throughout the rest of the hand, so you have the potential to gain more money from your opponents in future rounds of betting (and also you may have to pay more money to stay in the hand in later rounds of betting). In the example above, if you feel that your opponent will call a bet after the river, then if hit your flush you will be able to earn an additional $2. If you do not hit your flush you can fold the hand and not lose any additional money. So in this case your implied odds are 2/22, or 1 in 11, even better than your pot odds.
For all of the following problems we assume that all of the cards not in a player's hand or in the collection of community cards are drawn with equal probability. This is a valid assumption if we have no knowledge of the other players' cards (see the Blackjack lesson, problem 3 for further details). In actuality it may be possible to infer some information about an opponent's hand based on her betting patterns or behavior.
It is sometimes useful to know the frequency of each of the different poker hands. In Texas Hold'em, each player is making a hand out of seven available cards. To determine the probability of each hand occuring we can count the number of distinct ways of obtaining each hand and divide by the total number of possible hands. This requires thinking about all of the different ways of obtaining a given hand and coming up with an orderly process for counting these different ways. The easiest way to do this involves heavy use of combinations C_{n,m} (the number of ways of choosing m objects (in any order) from a collection of n objects. These are also called binomial coefficients. See the probability review for more details). For example, the total number of possible seven cards hands is equal to the number of ways of choosing seven distinct cards out of a collection of 52, giving a total of C_{52,7} = 133,784,560 hands. Note that even though two identical hands in different suits have the same value in poker they are being counted as distinct hands.
The difficulty of calculating these frequencies varies significantly by hand. The high ranking hands such as four of a kind and straight flush can only be obtained in a limited number of different ways and therefore it is therefore easier to calculate their frequencies. Try calculating these frequencies first. Once you get the hang these types of calculations, if you are up for a challenge you can attempt some of the more involved calculations.
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