Pot odds
Encyclopedia
In poker
, pot odds are the ratio of the current size of the pot
to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value
.
To convert any percentage or fraction to the equivalent odds, we subtract the numerator from the denominator and then divide this remainder by the numerator. For example, to convert 1/4 (or 25%), we subtract 1 from 4 to get a remainder of 3 (or 25 from 100 to get a remainder of 75) and then divide 3 by 1 (or 75 by 25), giving us 3, or exactly 3:1.
of that hand when the player is faced with a bet.
The expected value of a call is determined by comparing the pot odds to the odds of drawing a card that wins the pot. When the odds of drawing a card that wins the pot are numerically higher than the pot odds, the call has a positive expectation; on average, you win a portion of the pot that is greater than the cost of the call. Conversely, if the odds of drawing a winning card are numerically lower than the pot odds, the call has a negative expectation, and you can expect to win less money on average than it costs to call the bet.
with little chance of improving what he believes may currently be the best hand, but an opponent continues to bet. If the opponent has a weak hand, 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).
a made hand
that discourages opponents from chasing a drawing hand
.
for an opponent with a flush draw to call, but Bob does not 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 $10 (half the pot), when his opponent acts, the pot will be $30 and it will cost $10 to call. The opponent's pot odds will be 3-to-1, or 25 percent. If the opponent is on a flush draw (9/46, approximately 19.565 percent or 4.11-to-1 odds against 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 completes his flush draw (see implied pot odds).
A bet of $6.43, resulting in pot odds of 4.11-to-1, would make his opponent mathematically indifferent to calling if implied odds are disregarded.
, 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).
However, this conclusion does not take into account some of the context of specific situations. A player's bluffing frequency often accounts for many different factors, particularly the tightness or looseness of their opponents. Bluffing against a tight player is more likely to induce a fold than bluffing against a loose player, who is more likely to call the bluff. Sklansky's strategy is an equilibrium
strategy in the sense that it is optimal against someone playing an optimal strategy against it.
Poker
Poker is a family of card games that share betting rules and usually hand rankings. Poker games differ in how the cards are dealt, how hands may be formed, whether the high or low hand wins the pot in a showdown , limits on bet sizes, and how many rounds of betting are allowed.In most modern poker...
, pot odds are the ratio of the current size of the pot
Pot (poker)
The pot in poker refers to the sum of money that players wager during a single hand or game, according to the betting rules of the variant being played...
to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
.
Converting odds ratios to and from percentages
Odds are most commonly expressed as ratios, but converting them to percentages will often make them easier to work with. The ratio has two numbers: the size of the pot and the cost of the call. To convert this ratio to the equivalent percentage, we add these two numbers together and then divide the cost of the call by this sum. For example, the pot is $30, and the cost of the call is $10. The pot odds in this situation are 30:10, or 3:1 when simplified. To get the percentage, we add $30 and $10 to get a sum of $40 and then divide $10 by $40, giving us 1/4, or 25%.To convert any percentage or fraction to the equivalent odds, we subtract the numerator from the denominator and then divide this remainder by the numerator. For example, to convert 1/4 (or 25%), we subtract 1 from 4 to get a remainder of 3 (or 25 from 100 to get a remainder of 75) and then divide 3 by 1 (or 75 by 25), giving us 3, or exactly 3:1.
Using pot odds to determine expected value
When a player holds a drawing hand, or a hand that is behind now but is likely to win if a certain card is drawn, pot odds are used to determine the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of that hand when the player is faced with a bet.
The expected value of a call is determined by comparing the pot odds to the odds of drawing a card that wins the pot. When the odds of drawing a card that wins the pot are numerically higher than the pot odds, the call has a positive expectation; on average, you win a portion of the pot that is greater than the cost of the call. Conversely, if the odds of drawing a winning card are numerically lower than the pot odds, the call has a negative expectation, and you can expect to win less money on average than it costs to call the bet.
Implied pot odds
Implied pot odds, or simply implied odds, are calculated the same way as pot odds, but take into consideration estimated future betting. Implied odds are calculated in situations where the player expects to fold in the following round if the draw is missed, thereby losing no additional bets, but expects to gain additional bets when the draw is made. Since the player expects to always gain additional bets in later rounds when the draw is made, and never lose any additional bets when the draw is missed, the extra bets that the player expects to gain, excluding his own, can fairly be added to the current size of the pot. This adjusted pot value is known as the implied pot.Example (Texas Hold'em)
On the turn, Alice's hand is certainly behind, and she faces a $1 call to win a $10 pot against a single opponent. There are four cards remaining in the deck that make her hand a certain winner. Her probability of drawing one of those cards is therefore 4/46 (8.7%), which when converted to odds is 10.5:1. Since the pot lays 10:1 (9.1%), Alice will on average lose money by calling if there is no future betting. However, she expects her opponent to call her additional $1 bet on the last betting round if she makes her draw. She will fold if she misses her draw and thus lose no additional bets. Her implied pot is therefore $11 ($10 plus the expected $1 call to her additional $1 bet), so her implied pot odds are 11:1 (8.3%). Her call now has a positive expectation.Reverse implied pot odds
Reverse implied pot odds, or simply 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 implied 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 handMade hand
In poker, a made hand is one that does not need improvement to win, in contrast to a drawing hand. For example in draw poker, if you have two pairs, and your opponent is drawing for a straight or flush, you are said to have a made hand because even though you will be drawing a card just as he...
with little chance of improving what he believes may currently be the best hand, but an opponent continues to bet. If the opponent has a weak hand, 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 has a weak hand or is 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 plus $10 call on the river). Because she is risking $20 to win $30, Alice's reverse implied pot odds are 1.5-to-1 ($30/$20) or 40 percent (1/(1.5+1)). For calling to have a positive expectation, Alice must believe the probability of her opponent having a weak hand is over 40 percent.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 protectProtection (poker)
Protection in poker is a bet made with a strong but vulnerable hand, such as top pair when straight or flush draws are possible. The bet forces opponents with draws to either call with insufficient pot odds, or to fold, both of which are profitable for the betting player...
a made hand
Made hand
In poker, a made hand is one that does not need improvement to win, in contrast to a drawing hand. For example in draw poker, if you have two pairs, and your opponent is drawing for a straight or flush, you are said to have a made hand because even though you will be drawing a card just as he...
that discourages opponents from chasing a drawing hand
Draw (poker)
A poker player is drawing if they have a hand that is incomplete and needs further cards to become valuable. The hand itself is called a draw or drawing hand. For example, in seven-card stud, if four of a player's first five cards are all spades, but the hand is otherwise weak, they are drawing to...
.
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 wrongFundamental theorem of poker
The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information....
for an opponent with a flush draw to call, but Bob does not 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 $10 (half the pot), when his opponent acts, the pot will be $30 and it will cost $10 to call. The opponent's pot odds will be 3-to-1, or 25 percent. If the opponent is on a flush draw (9/46, approximately 19.565 percent or 4.11-to-1 odds against 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 completes his flush draw (see implied pot odds).
A bet of $6.43, resulting in pot odds of 4.11-to-1, would make his opponent mathematically indifferent to calling if implied odds are disregarded.
Bluffing frequency
According to David SklanskyDavid Sklansky
-Life and career:Sklansky was born and raised in Teaneck, New Jersey, where he graduated from Teaneck High School in 1966. He attended the University of Pennsylvania, but left before graduation. He returned to Teaneck and passed multiple Society of Actuaries exams by the time he was 20, and worked...
, game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
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
Value (poker)
In poker, the strength of a hand is often called its value; however, in the context of poker strategy the term is more often used to describe a betting tactic, a bet for value. This bet is intended to increase the size of the pot, by inducing opponents to call...
(one out of three times).
However, this conclusion does not take into account some of the context of specific situations. A player's bluffing frequency often accounts for many different factors, particularly the tightness or looseness of their opponents. Bluffing against a tight player is more likely to induce a fold than bluffing against a loose player, who is more likely to call the bluff. Sklansky's strategy is an equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
strategy in the sense that it is optimal against someone playing an optimal strategy against it.
See also
- List of poker terms
- Poker strategyPoker strategyPoker is a popular card game that combines elements of chance and strategy. There are various styles of poker, all of which share an objective of presenting the least probable or highest-scoring hand. A poker hand is a configuration of five cards, either held entirely by a player or drawn partly...
- Poker probabilityPoker probabilityIn poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.-Frequency of 5-card poker hands:...
- Poker probability (Texas hold 'em)Poker probability (Texas hold 'em)In poker, the probability of many events can be determined by direct calculation. This article discusses computing probabilities for many commonly occurring events in the game of Texas hold 'em and provides some probabilities and odds for specific situations...
- Poker probability (Omaha)Poker probability (Omaha)In poker, the probability of many events can be determined by direct calculation. This article discusses how to compute the probabilities for many commonly occurring events in the game of Omaha hold 'em and provides some probabilities and odds for specific situations...