What Are The Odds Of Different Poker Hands
This is a discussion on Are the Odds Different Online VS Live? Within the online poker forums, in the General Poker section; I am a fun play poker player and would like to know the difference.
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Introduction
Rules
- A single 52-card deck is used. All cards count as its poker value. Aces may be high or low.
- One player is designated as the dealer, usually with a laminated marker. This person does not have to physically deal the game. However it is important that a symbolic dealer position rotate around the table.
- The player to the dealer's left must make a 'small blind' bet. The player to the left of the small blind must make a 'big blind' bet. The amounts of both blinds should be specified in advance. The purpose of the blinds is to get the ball rolling with some money in the pot.
- Two cards shall be dealt down to each player, starting with the person to the dealer's left.
- The player to the left of the big blind must either call or raise the big blind bet. The play in turn will go around the table according to normal poker rules, which I assume the reader already knows. Table rules will specify any limits on the size or number of allowed raises.
- The small blind may also raise the big blind. If nobody raises the big blind the player making the big blind has the option to raise his own bet. The term for this is the 'big blind option.'
- Three community cards will be dealt face up in the center of the table. This is called the 'flop.'
- Another round of betting will ensue, starting with the player to the dealer's left.
- A fourth community card will be dealt face up in the center of the table. This card is called the 'turn.'
- Another round of betting will ensue, starting with the player to the dealer's left. Generally the minimum bet is double the first two rounds of betting.
- A fifth and final community card will be dealt face up in the center of the table. This card is called the 'river.'
- Another round of betting will ensue, starting with the player to the dealer's left. The minimum bet is generally the same as the previous round.
Each player still in the game at the end will determine the highest poker value among his own two cards and the five community cards. It is NOT a requirement that the player use both of his own cards. The player with the hand of highest poker value shall win. Following are the hand rankings.
- Straight flush: Five consecutive and suited cards. For example 5, 6, 7, 8, 9.
- Four of a kind: Four cards of the same rank, plus any fifth card. For example Q, Q, Q, Q ,4.
- Full house: Three of a kind and a pair. For example 6, 6, 6, J , J.
- Flush: Any five cards of the same suit, except for a higher ranking straight flush. For example A, Q, 8, 4 , 3.
- Straight: Five consecutive cards, except for a higher ranking straight flush. For example 8, 9, 10, J, Q.
- Three of a kind: Three cards of the same rank, plus any other two cards. For example 5, 5, 5, Q ,2 .
- Two pair: Two pairs, plus any fifth card. For example 8, 8, 2, 2 ,Q .
- Pair: A pair and any other three cards. For example 7, 7, 2, 5 ,A .
- ? High: Any five cards that do not form any higher poker hand. A king high hand for example might be K, Q, 7, 5 ,4 .
- If two or more players have poker values of the same rank then the individual cards will be used to break the tie. If necessary all five cards will be considered.
- I get asked a lot whether the two unused cards in a player's hand are used to break a tie. The answer is a firm NO. The two unused cards do not matter.
- If a new player arrives at the table he should either wait for the big blind position or put up an amount equal to the big blind, amounting to a call of the big blind.
- If a bet is made after another player runs out of money, then a separate pot is created. The player that ran out of money is not eligible to win the second pot. If more than one player runs out of money then multiple separate pots can be created.
- In formal games players may not bet with cash or buy chips with cash in the middle of a hand.
- There are numerous rules of etiquette, which I won't get into.
- There house may set the betting rules. There are three main types. A 'structured' game features raises of specified amounts. For example a '3/6 game' would mean that raises after the deal and flop are $3, and after the turn and river are $6. There is usually a limit to the number of raises a player may make, typically three. A 'pot limit' game has structured minimum raises but the maximum raise may be anything up to the amount in the pot at the time the raise is made. A 'no limit' game also has structured minimum raises but there is no maximum raise.
Examples
Example 1
Board: A, 2, 4, 5, 6
Player 1: J, 6
Player 2: 7, Q
Player 1 wins. Both have an ace high flush, so the second highest card is considered. Player 1's jack beats player 2's 7. The only way to have a flush tie is if the flush is entirely on the board and no hole cards are higher than the lowest card on the board in the same suit.
Example 2
Board: J, A, 7, 5, 6
Player 1: 2, J
Player 2: 10, J
Player 2 wins. Both have a pair of jacks so the singletons are considered. High highet singleton in both hands is an ace so the second highest singleton is considered. Player 1's second highest singleton is a 7, compared to player 2's 10. A 10 beats a 7 so player 2 wins.
Example 3
Board: A, A, K, Q, J
Player 1: Q, J
Player 2: Q, 2
Tie. Both have a two pair of aces and queens, with a king singleton. Some people incorrectly believe that in such cases the unused cards are considered, in this case player 1's pair of jacks beating player 2's jack/2. Only the top five cards matter. The jacks and deuce are irrelevant.
One of the most important aspects of Texas Hold'em is the value of each two-card hand before the flop. The decision of how to play your first two cards is something you face every hand, and the value of your first two cards is highly correlated to your probability of winning.
The following table shows my power rating for each initial 2-card hand in a 10-player game. The numbers are on a 0 to 40 scale. Basically, you should only play hands that are dark green, blue, or purple. Of course you should be more be more liberal in late position and picky in early position. If forced I would say you should need 10 points in late position and 19 points in early position to call the big blind. If your table is loose, as if often the case online, you can play a bit looser yourself.
Use the top table if you have a pair, the middle table if your cards are suited, and the bottom table if your cards are unsuited. Except for a pair,look up your high card along the left and your low card along the top.
Following are the links to my tables of the value of each intial hand according to the number of players. The 10-player section explains the methodology for creating the table table.
Pot Odds
The following table shows the probability of making various hands after the flop and the correct 'pot odds.' The pot odds are the breakeven ratio of money in the pot to the amount you have to bet for the player to be indifferent about calling, assuming the player would definitely win if he makes the hand (a big if) and there are no additional bets (another big if). This table is a good starting point the player should make mental adjustments for the probability of winning without making the hand, losing with making the hand, and expected future bets. The odds of a two pair improving to a full house are the same as those for four to an inside straight.
Pot Odds — After Flop
| Hand | Probability of Making Hand | Pot Odds |
|---|---|---|
| Four to a flush | 34.97% | 1.86 |
| Four to an outside straight | 31.45% | 2.18 |
| Four to an inside straight | 16.47% | 5.07 |
The next table shows the pot odds after the turn.
Pot Odds — After Turn
| Hand | Probability of Making Hand | Pot Odds |
|---|---|---|
| 4 to a flush | 19.57% | 4.11 |
| 4 to an outside straight | 17.39% | 4.75 |
| 4 to an inside straight | 8.70% | 10.50 |
Hand Strength Calculator
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Internal Links
- Pinapple — Strategy and analysis of which card to discard before the flop.
- Bad Beat Jackpots: What is the Probability of Hitting one?
- Texas Hold 'Em Dominated Hand Probabilities: What is the probability one of your opponents has similar, and better, hole cards than yours?
POKER PROBABILITIES
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5 Card Poker probabilities
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Ranking of poker hands
In 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
The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)
The nCr function on most scientific calculators can be used to calculate hand frequencies; entering nCr with 52 and 5, for example, yields as above.
| Hand | Frequency | Approx. Probability | Approx. Cumulative | Approx. Odds | Mathematical expression of absolute frequency |
|---|---|---|---|---|---|
| Royal flush | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
| Straight flush (excluding royal flush) | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
| Four of a kind | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
| Full house | 3,744 | 0.144% | 0.170% | 693.2 : 1 | |
| Flush (excluding royal flush and straight flush) | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
| Straight (excluding royal flush and straight flush) | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
| Three of a kind | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
| Two pair | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
| One pair | 1,098,240 | 42.3% | 49.9% | 1.36 : 1 | |
| No pair / High card | 1,302,540 | 50.1% | 100% | .995 : 1 | |
| Total | 2,598,960 | 100% | 100% | 1 : 1 |
The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.
When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.
Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.
The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.
Derivation of frequencies of 5-card poker hands
of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).
- Straight flush — Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (10-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is:
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
- or simply . Note: this means that the total number of non-Royal straight flushes is 36.
- Royal straight flush — A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits). Thus, the total number of royal straight flushes is
- Four of a kind — Any one of the thirteen ranks can form the four of a kind by selecting all four of the suits in that rank. The final card can have any one of the twelve remaining ranks, and any suit. Thus, the total number of four-of-a-kinds is:
- Full house — The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and consists of three of the four suits. The pair can be any one of the remaining twelve ranks, and consists of two of the four suits. Thus, the total number of full houses is:
Funny What Are The Odds
- Flush — The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:
- Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-10. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:
- Three of a kind — Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The remaining two cards can have any two of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of three-of-a-kinds is:
- Two pair — The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:
- Pair — The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is:
- No pair — A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is:
- Any five card poker hand — The total number of five card hands that can be drawn from a deck of cards is found using a combination selecting five cards, in any order where n refers to the number of items that can be selected and r to the sample size; the '!' is the factorial operator:
This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.
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