How To Calculate The Odds Of Hitting A Flush
5 June 2009Knowing the odds in a texas holdem poker game is very important. Knowing the correct poker odds will help you make better decisions, and better decisions in poker lead to more money in your pocket. In addition to just knowing poker odds, it is also useful to know how the odds are calculated. It will give you a better understanding of the game, and you might want to calculate some odds that you can’t just look up. This page will show you how to calculate the odds of hitting a flush, for any suited or unsuited cards, and how the chances of hitting your flush change from preflop odds depending on what cards come or don’t come. If you just want to see a table of the odds, and don’t care how they were calculated, go to the bottom of this page.
A Brief Summary On How Odds Are Calculated
In poker, if we want to find the probability of hitting your hand, all you need to do is calculate how many hand combinations there are that will hit your hand, and divide by the total hand combinations possible. You then know the percent chance of a certain event (a flush for instance) happening. The tool that we will use to help us find the total number of hands that help us, and the total number of hands possible is combinatorics.
In Poker, Card Order Is Not Important
We can use combinatorics to calculate odds because the order that the cards come is not important. The card order is important only if it affects the betting. Thus a card coming on the river is different than it coming on the flop, but a card first off the flop is the same as 3rd off the flop. If you get dealt a hand of Ace of spades, King of Spades, it is exactly the same as the hand King of Spades, Ace of Spades. On the flop, it doesn’t matter if the cards come out 5-6-9 or 6-9-5. So all we care about is the combinations of cards that come out, not what order they are in. (The order of the cards would be permutations)
How To Calculate Combinations
To calculate the number of possible poker hand combinations you need to know the number of starting cards, and how many you are selecting from it. If you want to know the number of unique 2 card hands you would calculate the number of 2 card combinations you can make out of 52 cards to choose from. From here on in we will denote this as Comb(52,2).
| Equation For Combinations | ||
| Description | Equation | Result |
| Calculate the number of 2 card combinations in 52 possibilities | (52 * 51 ) / ( 1 * 2) | 1326 |
| Calculate the # of 3 card combinations out of 50 cards, Comb(50,3) | ( 50*49*48 ) / (1*2*3) | 19600 |
| The number of different combinations of aces in a 2 card hand, Comb(4,2) | ( 4 * 3 ) / (2 * 1) | 6 |
You can check that for yourself by dealing it out. The ace combinations are AsAh, AsAd, AsAc, AhAd, AhAc, AdAc
Once you understand combinations there is really nothing to them. Applying them to poker hands will be the fun part.
Odds of Getting Suited Cards
Let’s start by calculating the chances of getting suited cards in the hole. We already calculated that the number of possible hole card combinations are Comb(52,2) = 1326. So how many of them are suited ?
If we start with an ace of any suit there are 12 suited combinations with the ace as the high card. AK, AQ, AJ … A3, A2. There are 11 suited combinations with the King as the high card KQ, KJ, KT, … K3, K2. There are 10 with Queen High, 9 Jack High, going all the way down to 1 with 3 high (32). Adding those = 12+11+10+9+8…+3+2+1 = 78. But that was just one suit. Since there are 4 suits the total number of suited possibilities are 78*4 = 312
So 312 suited combinations / 1326 possibles = .235
The odds of suited cards in the hole are 23.5%
For every 1 suited combination there are 3 unsuited combinations. For instance for every Ace-King of spades, you will have an Ace-spades King-hearts, Ace-spades King-clubs, Ace-spades King-diamonds. So if there are 3 times as many unsuited combinations as suited ones then 3 * 312 = 936
936 unsuited / 1326 total = 70.6% of getting an unsuited card in the hole
Since you have odds of drawing a pair in the hole 5.9% of the time, we can check the calculations by 23.5% + 70.6% + 5.9% = 100% chance that you either have suited cards, unsuited cards, or a pocket pair in the hole. ( 100% chance as long as you are playing Texas holdem….)
What Are The Odds of Hitting A Flush ?
Whatever hand you have you will be interested in your odds of winning at a showdown. So after all 5 cards have come out, what are your chances to hit your flush ?
Assuming you started with suited cards in the hole you calculate the preflop odds of hitting a flush by the river as such.
If you have 2 hearts as your hole cards, you either need 3 hearts to come on the board, 4 hearts to come on the board, or 5 hearts to come on the board. (We’re ignoring the chances of 5 spades, clubs, or diamonds coming if you hold hearts, because then the best you can do is split the pot. If 5 hearts come to the board, we’re hoping you aren’t holding 2 low hearts in your hand so you don’t just play the board)
To start, how many possible 5 card board combinations are there? Well you already know 2 cards, your two hole cards, so there are only 50 unknown cards left in the deck.
| Odds Of Hitting A Flush With Suited Cards | ||
| Description | Equation | Result |
| Possible 5 Card Board Combinations - 2 Hold Cards Are Known, So 50 Unknown are left, Comb(50,5) | (50*49*48*47*46) / (1*2*3*4*5) | 2,118,760 board combinations |
| How Many Hit The Flush ? | ||
| 5 Flush Cards On The Board | Comb(11,5) | 462 |
| 4 Flush Cards On The Board | Comb(11,4) * 39 (number of non flush cards in deck) | 12870 |
| 3 Flush Cards On The Board | Comb(11,3) * Comb(39,2) (any combination of the 39 no hearts left) | 122265 |
| Total Flush Possibilities | 462+12870+122265 | 135597 |
| Flush Odds | 135,597 / 2,118,760 | 0.0639 |
Suited cards have a 6.4% chance of hitting a flush by the river! The 6.4% chance of getting your suited cards to flush adds expected value to suited hole cards, but not as much as some people seem to think.
What About Unsuited Cards ?
Unsuited cards can only hit a flush if there are 4 or 5 flush cards on the board. But they can do it for either of the 2 suits of the unsuited hole cards. In addition, if you have a Heart and a Spade in the hole, there would be 12 hearts and 12 spades left in the deck, instead of 11 left in the deck if you have suited hole cards.
| Odds Of Hitting A Flush With Unsuited Cards | ||
| Description | Equation | Result |
| # Of Possible 5 Flushes | 2 Suits * Comb(12,5) | 1584 |
| # Of Possible 4 Flushes | 2 Suits * Comb(12,4) * 39 unsuited cards | 38610 |
| Total Odds | ( 1584 + 38610 ) / 2,118760 | 0.0189 |
Unsuited cards have a 1.9% chance of hitting a flush by the river. The probability of getting a flush with suited cards is 4.5% greater than the chances to hit a flush with unsuited hole cards. Of course, that doesn’t mean that any pair of suited cards are worth playing ….
But Wait, There’s More !
We can also calculate the probability of flopping a flush, or flopping a flush draw, as opposed to hitting the flush by the river. For this we will only work with suited hole cards, although you can use the same procedure to calculate the odds with unsuited cards. (Note, I did all the calculations on this page with Google Toolbar, you don’t need any kind of fancy programs or calculators to do poker math)
How many possible flops are there? Well you know 2 cards, your 2 hearts in the hole, so we want any 3 card combination out of 50 cards.
| Flush Odds On The Flop | ||
| Description | Equation | Result |
| Possible Flops, 50 Unknown Cards | Comb(50,3) | 19600 |
| Chances To Flop A Flush, 3 hearts out of 11 left | Comb(11,3) | 165 |
| Chances To Flop A Flush Draw, 2 hearts out of 11 left | Comb(11,2) * 39 | 2145 |
| Backdoor Flush Draw, 11 hearts left, 39 unsuited cards in any combination of 2 | 11 * Comb(39,2) | 8151 |
| Flopped Flush Odds | 165 / 19,600 | 0.0084 |
| Flopped 4 Flush Draw | 2145 / 19,600 | 0.1094 |
| Flopped Backdoor Draw | 8151 / 19,600 | 0.4159 |
If you have suited hole cards, your odds of hitting the flop with a made flush is .84%. You will flop a 4 card flush draw 10.94% of the time, and a backdoor flush draw 41.6 % of the time.
Finally, what are the odds of hitting a flush draw ?
After the flop, you can see 5 cards, so there are 47 cards which could be dealt. If you have a 4 flush there are 9 flush cards left. You can either hit a flush card, non flush card, or a flush card, flush card. The number of 2 card combinations out of 47 are Comb(47,2) = 1081
| Odds Of Hitting A Flush Draw | ||
| Description | Equation | Result |
| Possible Turn-River Combinations, 47 Unknown Cards | Comb(47,2) | 1081 |
| Hit A 4 Flush | ||
| Turn-River = Flush-Flush Cards (9 flush cards left) | Comb(9,2) | 36 |
| Turn-River = Flush-Non Flush (9 flush cards, 38 Non Flush) | 9 * 38 | 342 |
| Odds Of Hitting A 4 Flush By The River | ( 36 + 342) / 1081 | 0.349 |
| Hitting A Backdoor Flush Draw | ||
| Need 2 Running Flush Cards Out Of 10 | Comb(10,2) | 45 |
| Backdoor Flush Odds | 45 / 1081 | 0.0416 |
If you have a 4 card flush draw, you have a 34.9 % chance of making your flush by the river. Your odds of hitting a backdoor flush draw are 4.16%.
| Comprehensive Texas Holdem Flush Odds | |
| Preflop | |
| Probability Of Getting Suited Cards | 23.50% |
| Probability Of Getting Unsuited Cards | 70.60% |
| On The Flop w/ Suited Hole Cards | |
| Odds Of Hitting A Flush | 0.84% |
| Odds Of Getting a 4 Card Flush Draw | 10.90% |
| Odds Of Getting 3 Card (backdoor) Flush Draw | 41.60% |
| Drawing | |
| Odds of Hitting Your 4 Card Flush Draw By The River | 34.90% |
| Odds of Hitting Your Backdoor Flush Draw By The River | 4.16% |
| Total Odds, By The River | |
| Chances For of Suited Cards To Hit A Flush | 6.40% |
| Chances For of Unsuited Hole Cards To Hit A Flush | 1.90% |
No comments yet