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Poker players use pot odds to determine the expected value (profitability over the long run) of a play . In general, odds may be expressed as a win-to-loss ratio. Odds may be converted into percentage probabilities using the formula: win-to-loss odds = win / (win + loss) % probability. For example 1-to-4 odds translate to 1 / (1 + 4) = 20% probability. Odds are also commonly expressed in terms of odds against (loss-to-win ratio). As a convention, this article uses odds for (win-to-loss).

For every potential action (check, fold, call, raise) at every point in a game of poker, the correct strategy is influenced by the pot odds facing the player (and offered to the opponent(s)). The lower the pot odds facing a call, the more likely it is that folding will be the correct play, and the higher the pot odds facing a call, the more likely it is that calling is the correct play. For example, if a player can call for $1 with a $1000 pot, there is essentially no hand that would be correct to fold, because the player only has to win one time in a thousand for the call to be profitable.

The probability or winning is the chance that the player's hand will win either by being the best hand at the showdown or because the opponents fold.

Texas hold 'em example

In Texas hold 'em, the approximate percentage probability that a player will hit an out on the next card is calculated as: (number of outs) x 2 + 1. For example, if a player has a potential flush and therefore 9 cards could improve his hand, there is roughly a 19% (9 x 2 + 1) probability the next card will give him his flush. With two cards to come, the approximate percentage probability is: (number of outs) x 4 - 1. See discussion of Poker probability (Texas hold 'em) for more details.

For an action to have a positive expectation, a player's odds of winning must be at least equal to the applicable pot odds.

Simple pot odds

Simple pot odds, or expressed pot odds, apply when considering a call when no further betting will be made (e.g., calling a bet on the final round). Simple pot odds are the ratio of the size of the potential bet to the size of the pot (bet-to-pot ratio). For example, if a player must call a $10 bet for a chance to win a $40 pot (not including the player's $10 call), the player's simple pot odds are 1-to-4 (20% probability). Continuing the example, assume the player's odds of winning are also 1-to-4. If the pot is played five times, the player puts in $10 five times, loses four times and wins $50 once (breaking even).

Simple pot odds apply on any betting round when making a pure bluff if the bluff will be given up if called or raised.

Implied pot odds

Implied pot odds, or implied odds, apply in situations where future betting may occur (e.g., with more cards or more draws to come) and the player's hand is currently a certain loser but may improve to a certain winner (e.g., improving from no pair to a nut flush). Precise calculation of implied odds for hands that may be probable winners is significantly more complex and not well-documented in poker literature. Note on terminology: some authors use the term implied pot odds to specifically refer to situations with one card (or draw) to come and the term effective implied pot odds to refer to situations with more than one card (or draw) to come.

A player's implied pot is the current pot plus the value of future bets expected from opponents that may be won, excluding the player's own bets. When figuring the implied pot, a player must estimate the bets expected from opponents in the event the player wins the pot.

Texas hold 'em example

Alice holds the A♠ and the board shows three low spades with one card to come. Alice believes she will only win if another spade comes on the river to make her a nut flush. To figure her implied pot, Alice must estimate the expected bets by her opponents if the spade comes on the last card. In that event, because of the fair chance the opponent may not have a high spade, Alice may reasonably have a low expectation of further contributions to the pot by her opponents.

A player's effective bet is the sum of the current potential bet plus all future bets a player expects to make to see the last card, excluding any bets on the end. When there is only one card to come, the effective bet is simply the current potential bet under consideration. A player's implied pot odds are the ratio of the effective bet to the implied pot.

Texas hold'em example, two cards to come

With two cards to come, Alice holds a nut flush draw after the flop and faces a $5 call to win a $20 pot. If Alice makes her flush, she expects her opponent to contribute another $10 on the turn and $10 on the river. Alice's effective call is $15 ($5 on the flop + $10 on the turn). Alice's implied pot is $40 ($20 current pot + $10 turn + $10 river). Alice's implied pot odds are $15-to-$40 or 27% (15 / (15 + 40)). A call by Alice has a positive expectation because the probability of making her flush (35% with two cards to come) is greater than the implied pot odds (27%).

Texas hold'em example, one card to come

With one card to come, Alice still holds a nut flush draw and faces a $10 call to win a $35 pot. If Alice makes her flush, she expects her opponent to contribute another $10 in the final round. Alice's implied pot is $45 ($35 current pot + 10 future bets by her opponent). Alice's implied pot odds are $10-to-$45 or 18% (10 / (10 + 45)). A call by Alice has an about break-even expectation because the probability of making her flush (19% with one card to come) is about the same as her implied pot odds (18%).

Reverse implied pot odds

Reverse implied pot odds, or reverse implied odds, apply to situations where a player will win the minimum if he has the best hand but lose the maximum if he does not have the best hand. Aggressive actions (bets and raises) are subject to reverse pot odds, because they win the minimum if they win immediately (the current pot), but may lose the maximum if called (the current pot plus the called bet or raise). These situations may also occur when a player has a made hand with little chance of improving which he believes may currently be the best hand, but an opponent continues to bet. If the opponent is weak or bluffing, he will likely give up after the player calls and not call any bets the player makes. If the opponent has a superior hand, he will continue the hand (extracting additional bets or calls from the player).

Limit Texas hold'em example

With one card to come, Alice holds a made hand with little chance of improving and faces a $10 call to win a $30 pot. If her opponent is weak or bluffing, Alice expects no further bets or calls from her opponent. If her opponent has a superior hand, Alice expects the opponent to bet another $10 on the end. Therefore, if Alice wins, she only expects to win the $30 currently in the pot, but if she loses, she expects to lose $20 ($10 call on the turn + $10 call on the river). Because she is risking $20 to win $30, Alice's reverse implied pot odds are $20-to-$30 or 40% (20 / (20 + 30)). For calling to have a positive expectation, Alice must believe her probability of winning the pot is at least 40%.

Manipulating pot odds

Often a player will bet to manipulate the pot odds offered to other players. A common example of manipulating pot odds is make a bet to protect a made hand that discourages opponents from chasing a drawing hand.

No-limit Texas hold 'em example

With one card to come, Bob has a made hand, but the board shows a potential flush draw. Bob wants to bet enough to make it wrong for an opponent with a flush draw to call, but Bob doesn't want to bet more than he has to in the event the opponent already has him beat. How much should Bob bet?

Assume a $20 pot and one opponent. If Bob bets half the pot ($10), the opponent faces a $10 call to win a $30 pot. The opponent's pot odds will be $10 call-to-$30 pot or 25% (10 / (10 + 30)). If the opponent is on a flush draw (19% with one card to come), the pot is not offering adequate pot odds for the opponent to call unless the opponent thinks he can induce additional final round betting from Bob if the opponent make his hand (see implied pot odds).

Bluffing frequency

Game theory shows that a player should bluff a percentage of the time equal to his opponent's pot odds to call the bluff. For example, in the final betting round, if the pot is $30 and a player is contemplating a $30 bet (which will give his opponent 2-to-1 pot odds for the call), the player should bluff half as often as he would bet for value (one out of three times).

See the article on bluffing for more details.

References

David Sklansky (1987). The Theory of Poker. Two Plus Two Publications. ISBN 1880685000.
David Sklansky (2001). Tournament Poker for Advanced Players. Two Plus Two Publications. ISBN 1880685280.
David Sklansky and Mason Malmuth (1988). Hold 'em Poker for Advanced Players. Two Plus Two Publications. ISBN 1880685221.
Dan Harrington and Bill Robertie (2004). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume I: Strategic Play. Two Plus Two Publications. ISBN 1880685337.
Dan Harrington and Bill Robertie (2005). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume II: The Endgame. Two Plus Two Publications. ISBN 1880685353.

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