Poker Math & Probabilities (Texas Hold'em). 15-to-1 6.50% Straight & Flush Draw Straight / Flush or Better 15 Three of a Kind Hope to Make Full House. Two suited cards will flop a flush Two suited cards will make a flush by the river The Probability that. Straight Pair Straight.
The odds of flopping a straight flush with a premium suited connector such as T9s is 0.02% or 1 in 4,900
Definition of the Straight Flush –
Five cards of consecutive rank, all of the same suit.
Example – 5d6d7d8d9dThe Ten to Ace Straight Flush is the strongest hand in poker and is referred to as the “Royal Flush”.
Odds of Making a Straight Flush on the Flop
Flopping a Straight Flush seldom happens in poker. We specifically need to start out with two suited connected cards for this to be possible.
The odds of flopping a Straight Flush are so unlikely (0.02% or less) that the majority of poker equity calculators don’t even show the precise odds.
We’ll need to do some maths of our own.
Calculation of Straight Flush Odds
Let’s start with a very specific example -
We hold A2s. What are the odds of flopping the Ace to Five Straight Flush?
Why do we choose this example? It’s the easiest one because it provides only one way of making the Straight Flush. The flop has to come down precisely Three, Four, Five of the correct suit.
So, how likely is this?
In order to calculate, we’ll first need to know how many combinations of three cards are possible on the flop.
Basic Combinations and Permutations
Firstly, how many different combinations of three cards can be dealt on the flop? Assuming we care about the order of the three cards (and that our two hole cards are already known), the answer is 117,600 (50 * 49 * 48).
In statistics, this type of calculation is referred to as a permutation and accounts for the order of the flop cards.
Of course, in Hold’em, the order of the cards on the flop doesn’t matter (i.e. a 3,4,5 flop is the same as a 5,3,4 flop, for all intents and purposes). What we are interested in is the number of possible combinations of three cards.
A combination is similar to a permutation but doesn’t account for the order. Since there are 6 possible ways of arranging three cards, we can simply divide our number of permutations (117,600) by 6 to establish the number of possible three-card combinations on the flop.
117,600 / 6 = 19,600 possible combinations of three cards on the flop (given two cards are known)
In other words, there are 19,600 different possible sets of three cards that may fall on the flop given that our two hole cards are already known.
Guess what?
To make the exact Straight Flush in question, only one of these 19,600 combinations will do the job.
Armed with that information, we can now establish a range of different probabilities.
Odds of flopping the Straight Flush with A2s = 1/19,600 = 0.00005 or roughly 0.005%
That’s an insanely small likelihood!
Thankfully, the odds with different types of starting hands are usually a little better.
It all depends on the number of different combinations of three cards that provide a Straight Flush.
For example, think about T9s.
How many different ways are there to make a Straight Flush with 9Ts?
Ways of making a Straight Flush with T9s
JQK
QJ8
J87
678
So that’s four different ways. We are hence four times as likely to make a Straight Flush with 9Ts as we are to make a Straight Flush with A2s.
Odds of flopping the Straight Flush with 9Ts = 4/19,600 = 0.0002 or roughly 0.02%
Ways of making a Straight Flush with T8s
QJ9
J79
679
Odds of flopping the Straight Flush with T8s = 3/19,600 = 0.00015 or roughly 0.015%
Ways of making a Straight Flush with T7s
J89
689
Odds of flopping the Straight Flush with T7s = 2/19,600 = 0.0001 or roughly 0.01%
Only suited connectors (or gappers) can make Straight Flushes on the flop. All other holdings such as pocket-pairs and off-suit combos can never flop a Straight Flush.
We are, naturally, more likely to flop a Straight Flush draw as opposed to the Straight Flush itself. To see examples of calculating the odds of hitting a Straight Flush draw on the flop, check out the 888poker article on Royal Flush odds in poker.
Odds of Making a Straight Flush on the Later Streets
There will be two primary types of Straight Flush draw we’ll flop. The gutshot Straight Flush draw and the open-ended Straight Flush draw.
Gutshot Straight Flush draws have 1 out in the deck, while open ended Straight Flush draws have 2 outs in the deck.
Odds of Hitting on the Turn or River
Odds of catching the gutshot Straight Flush on the turn 1/47 = 0.0213 or roughly 2.1%
Odds of catching the open-ended Straight Flush on the turn 2/47 = 0.426 or roughly 4.3%
Odds of catching the gutshot Straight Flush on the river 1/46 = 0.0217 or roughly 2.2%
Odds of catching the open-ended Straight Flush on the river 2/46 = 0.0435 or roughly 4.4%
Odds of Hitting by the River
To calculate the probability of hitting by the river, we’ll employ the trick of calculating the possibility of not hitting and then subtracting from 100%.
Odds of not catching the gutshot Straight Flush on the turn 46/47
Odds of not catching the open-ended Straight Flush on the turn 45/47
Odds of not catching the gutshot Straight Flush on the river 45/46
Odds of not catching the open-ended Straight Flush on the river 44/46
Odds of not catching the gutshot Straight Flush on the turn or river = 46/47 * 45/46 = 0.9574 or roughly 95.7%
Odds of not catching the open-ended Straight Flush on the turn or river = 45/47 * 44/46 = 0.9158 or roughly 91.6%
Odds of hitting the gutshot Straight Flush by the river = (100- 95.7%) roughly 4.3%
Odds of hitting the open-ended Straight Flush by the river = (100 – 91.6%) roughly 8.4%
Implied Odds Analysis of a Straight Flush
A Straight Flush always carries excellent implied odds when hitting. This is because our opponent is usually forced into stacking off with very strong worse hands such as worse flushes and full houses.
Straight Flushes made with two of our hole cards always carry better implied odds than Straight Flushes made with one of our hole cards.
When using just one of our hole cards, it means there will be four cards to the Straight Flush already on the board. This decreases the chance that our opponent will pay us off with worse holdings.
Although Straight Flushes should hardly ever be folded, their implied odds are the best when no higher Straight Flush is possible on the board.
Basic Strategy Advice
It’s basically the nuts. Play aggressively and make big bets! Even if a higher Straight Flush is possible, it’s usually just a cooler if we are beat. We’d have to be really deep to find an exception.
Odds of Making Straight Flush | |
Method (Straight Flush) | Probability (%) |
Flopping the Straight Flush with A2s | 0.01 |
Flopping the Straight Flush with T9s | 0.02 |
Flopping the Straight Flush with T8s | 0.02 |
Flopping the Straight Flush with T7s | 0.02 |
Catching the Straight Flush Gutshot from flop to turn | 2.13 |
Catching the Straight Flush open ender from flop to turn | 4.26 |
Catching the Straight Flush Gutshot from turn to river | 2.17 |
Catching the Straight Flush open ender from turn to river | 4.35 |
Catching the Straight Flush Gutshot from flop to river | 4.30 |
Catching the Straight Flush open ender from flop to river | 8.42 |
On This Page
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
What Are The Odds Of Flopping A Royal Flush In Texas Holdem
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Odds Of Flopping A Royal Flush In Texas Holdem
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Internal Links
What Are The Odds Of Flopping A Straight Flush In Texas Holdem
- 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?