Import:Pot odds

In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. In other words, if the pot contains $100, and a player must call $10 to stay in the hand, then the player has 100-to-10, or 10-to-1 (commonly expressed as 10:1), pot odds. 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. Indeed, a common usage of the term is to say that one "has pot odds", meaning that the present pot odds, compared to one's estimated chance of winning, make it profitable to call.

Converting pot odds to percentage values
Pot odds are a ratio, but having their percentage value 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 get the percentage value, we add the Size of the Pot and the Cost of the Call numbers together and record their Sum. We then divide the Cost of the Call by the Sum.

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 3 plus 1 get a sum of 4. Now we divide 1 by 4, giving us 0.25, or 25 percent.

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 value 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 second to last betting round, 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 odds of drawing to one of those cards is 10.5:1 (8.7 percent). Since the pot lays 10:1, Alice will lose money by calling if there is no future betting. However, she expects her opponent to call her additional $1 bet which she will make when she makes her draw. She will fold when she misses her draw (and lose no additional bets). Her implied pot odds are 11:1 ($10 plus the expected $1 call, to her additional $1 bet). This 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 hand with little chance of improving what 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 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 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 $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 percent or 4.3-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 $6 bet (or more precisely, $6.13), resulting in pot odds of 4.3-to-1, would make his opponent mathematically indifferent to calling.

Bluffing frequency
According to David Sklansky, 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 is a purely theoretical conclusion, which does not take into account the specifics of the situation. In reality, a player's bluffing frequency should account for many different factors, most importantly the tightness or looseness of their opponents. Bluffing against a tight player is much more profitable than bluffing against a loose player, who is more likely to call the bluff.