Joined: Mar '09
Location: Canada
Age: 38 (M)
Posts: 3211
heres a site i ran into a while back..been reading it and always look back to it while playing poker..most of us have already studied this but some of the newbies to poker may be interested!
this is all copy and paste!! did not write any myself lol it took me about 2hours to fix charts and cut stuff out so i really hope someone has some interest in this! hopefully atleast one will learn to make my efforts worth while! ( sorry for the charts hadda fix them up so a bit complicated)
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[1] for specific situations. In most cases, the probabilities and odds are approximations due to rounding.
When calculating probabilities for a card game such as Texas Hold 'em, there are two basic approaches. The first approach is to determine the number of outcomes that satisfy the condition being evaluated and divide this by the total number of possible outcomes. For example, there are six outcomes (ignoring order) for being dealt a pair of aces in Hold' em: {A♣, A♥}, {A♣, A♠}, {A♣, A♦}, {A♥, A♠}, {A♥, A♦} and {A♠, A♦}. There are 52 ways to pick the first card and 51 ways to pick the second card and two ways to order the two cards yielding (52×51)/2=1326 possible outcomes when being dealt two cards (also ignoring order). This gives a probability of being dealt two aces of .
The second approach is to use conditional probabilities, or in more complex situations, a decision tree. There are 4 ways to be dealt an ace out of 52 choices for the first card resulting in a probability of There are 3 ways of getting dealt an ace out of 51 choices on the second card after being dealt an ace on the first card for a probability of The conditional probability of being dealt two aces is the product of the two probabilities.
Here are the probabilities and odds of being dealt various other types of starting hands.
Hand Probability Odds AKs (or any specific suited cards) -----------331 : 1 AA (or any specific pair) ------------------- 220 : 1 AKs, KQs, QJs, or JTs (suited cards)-------- 81.9 : 1 AK (or any specific non-pair incl. suited) ----81.9 : 1 AA, KK, or QQ ----------------------------------72.7 : 1 AA, KK, QQ or JJ ------------------------------ 54.3 : 1 Suited cards, jack or better ------------------ 54.3 : 1 AA, KK, QQ, JJ, or TT -------------------------43.2 : 1 Suited cards, 10 or better --------------------32.2 : 1 Suited connectors ----------------------------- 24.5 : 1 Connected cards, 10 or better -------------- 19.7 : 1 Any 2 cards with rank at least queen ------- 19.1 : 1 Any 2 cards with rank at least jack----------- 10.1 : 1 Any 2 cards with rank at least 10 ------------ 5.98 : 1 Connected cards (cards of consecutive rank) 5.38 : 1 Any 2 cards with rank at least 9 ------------- 3.81 : 1 Not connected nor suited, at least one 2-9 - 0.873 : 1
possible head-to-head match ups in Hold 'em. (The total number of match ups is divided by the two ways that two hands can be distributed between two players to give the number of unique match ups.) However, since there are only 169 distinct starting hands, there are 169 × 1,225 = 207,025 distinct head-to-head match ups.[2]
It is useful and interesting to know how two starting hands compete against each other heads up before the flop. In other words, we assume that neither hand will fold, and we will see a showdown. This situation occurs quite often in no limit and tournament play. Also, studying these odds helps to demonstrate the concept of hand domination, which is important in all community card games.
This problem is considerably more complicated than determining the frequency of dealt hands. To see why, note that given both hands, there are 48 remaining unseen cards. Out of these 48 cards, we can choose any 5 to make a board. Thus, there are 1 712 304 possible boards that may fall. In addition to determining the precise number of boards that give a win to each player, we also must take into account boards which split the pot, and split the number of these boards between the players.
The problem is trivial for computers to solve by brute force search; there are many software programs available that will compute the odds in seconds. A somewhat less trivial exercise is an exhaustive analysis of all of the head-to-head match ups in Texas Hold 'em, which requires evaluating each possible board for each distinct head-to-head match up, or 1,712,304 × 207,025 = 354,489,735,600 (≈354 billion) results
Head-to-head starting hand matchups....
When comparing two starting hands, the head-to-head probability describes the likelihood of one hand beating the other after all of the cards have come out. Head-to-head probabilities vary slightly for each particular distinct starting hand matchup, but the approximate average probabilities, as given by Dan Harrington in Harrington on Hold'em [p. 125], are summarized in the following table.
Favorite-to-underdog matchup Probability Odds for Pair vs. 2 undercards -------------------- 4.9 : 1 Pair vs. lower pair ------------------------ 4.5 : 1 Pair vs. 1 overcard, 1 undercard ------- 2.5 : 1 2 overcards vs. 2 undercards ----------- 1.7 : 1 Pair vs. 2 overcards ---------------------- 1.2 : 1 These odds are general approximations only derived from averaging all of the hand matchups in each category. The actual head-to-head probabilities for any two starting hands vary depending on a number of factors, including:
Suited or unsuited starting hands; Shared suits between starting hands; Connectedness of non-pair starting hands; Proximity of card ranks between the starting hands (lowering straight potential); Proximity of card ranks toward A or 2 (lowering straight potential); Possibility of split pot. For example, A♠ A♣ vs. K♠ Q♣ is 87.65% to win (0.49% to split), but A♠ A♣ vs. 7♦ 6♦ is 76.81% to win (0.32% to split).
The following table shows the number of hand combinations for up to nine opponents.
Opponents Number of possible hand combinations 1 1,225 2 690,900 3 238,360,500 4 56,372,258,250 5 ≈9.7073 × 1012 (more than 9 trillion) 6 ≈1.2620 × 1015 (more than 1 quadrillion) 7 ≈1.2674 × 1017 (more than 126 quadrillion) 8 ≈9.9804 × 1018 (almost 10 quintillion) 9 ≈6.2211 × 1020 (more than 622 quintillion)
The following table shows the probability that before the flop another player has a larger pocket pair when there are one to nine other players in the hand.
Probability of facing a larger pair when holding anginst x amount of players
The value of a starting hand can change dramatically after the flop. Regardless of initial strength, any hand can flop the nuts—for example, if the flop comes with three 2s, any hand holding the fourth 2 has the nuts (though additional cards could still give another player a higher four of a kind or a straight flush). Conversely, the flop can undermine the perceived strength of any hand—a player holding A♣ A♥ would not be happy to see 8♠ 9♠ 10♠ on the flop because of the straight and flush possibilities.
there are 19 600 possible flops for any given starting hand. By the turn the total number of combinations has increased to 230 300 and on the river there are 2 116 760 possible boards to go with the hand.
The following are some general probabilities about what can occur on the board. These assume a "random" starting hand for the player.
Board consisting of Making on flop Making by turn Making by river flop ----------turn---------------river Three or more of same suit --- 18.3 : 1 ------- 6.40 : 1 --------- 3.24 : 1 Four or more of same suit ----------------------- 93.7 : -------------28.5 : 1 Rainbow flop (all different suits)- 1.51 : 1 ------8.48 : 1 Three cards of consecutive rank (but not four consecutive) -27.8 : 1 -- 7.46 : 1 ---2.99 : 1 Four cards to a straight (but not five)-----------24.8 : 1 ------------4.27 : 1 Three or more cards of consecutive rank and same suit 459 : 1-- 114 : 1 --- 45.0 : 1 Three of a kind (but not a full house or four of a kind) 424 : 1 -- 106 : 1 -- 46 : 1 A pair (but not two pair or three or four of a kind) 4.90 : 1 --2.29 : 1 ---- 1.36 : 1 Two pair (but not a full house) -----------------95.4 : 1 -- 20.2 : 1
The following table gives the probability that no overcards will come on the flop, turn and river, for each of the pocket pairs from 3 to K.
Notice that there is a better than 35% probability that an ace will come by the river if holding pocket kings, and with pocket queens, the odds are slightly in favor of an ace or a king coming by the turn, and a full 60% in favor of an overcard to the queen by the river. With pocket jacks, there's only a 43% chance that an overcard will not come on the flop and it is better than 3 : 1 that an overcard will come by the river.
Notice, though, that those probabilities would be lower if we consider that at least one opponent happens to hold one of those overcards.
The probabilities of drawing these outs are easily calculated. At the flop there remain 47 unseen cards, so the probability is (outs ÷ 47). At the turn there are 46 unseen cards so the probability is (outs ÷ 46). The cumulative probability of making a hand on either the turn or river can be determined as the complement of the odds of not making the hand on the turn and not on the river. The probability of not drawing an out is (47 − outs) ÷ 47 on the turn and (46 − outs) ÷ 46 on the river; taking the complement of these conditional probabilities gives the probability of drawing the out by the river.
outs(xnumber) turn 1st ------ river 2nd ------make on turn or river 3rd Inside straight flush; Four of a kind (1) 46.0 : 1----- 45.0 : 1 ----- 22.5 : 1 Open-ended straight flush; Three of a kind (2) 22.5 : 1----- 22.0 : 1------10.9 : 1 High pair (3) 14.7 : 1------ 14.3 : 1 ----- 7.01 : 1 Inside straight; Full house (4) 10.7 : 1 ----- 10.5 : 1 ------ 5.07 : 1 Three of a kind or two pair (5) 40 : 1 -----20 : 1 ------- 3.91 : 1 Either pair (6) 6.83 : 1 ----- 6.67 : 1 ------ 3.14 : 1 Full house or four of a kind; (see note) Inside straight or high pair (7) 5.71 : 1 ------ 5.57 : 1 ------ 2.59 : 1 Open-ended straight (8) 4.88 : 1 ------4.75 : 1------- 2.18 : 1 Flush (9) 4.22 : 1 ----- 4.11 : 1 -------- 1.86 : 1 Inside straight or pair (10) 3.70 : 1----- 3.60 : 1 ---- 1.60 : 1 Open-ended straight or high pair (11) 3.27 : 1 ----- 3.18 : 1 ------ 1.40 : 1 Open-ended straight or pair (14) 2.36 : 1 ----- 2.29 : 1 ------ 0.955 : 1
Note: When drawing to a full house or four of a kind with a pocket pair that has hit trips (three of a kind) on the flop, there are 6 outs to get a full house by pairing the board and one out to make four of a kind. This means that if the turn does not pair the board or make four of a kind, there will be 3 additional outs on the river, for a total of 10, to pair the turn card and make a full house. This makes the probability of drawing to a full house or four of a kind on the turn or river 0.334 and the odds are 1.99 : 1. This makes drawing to a full house or four of a kind by the river about 8½ outs.
If a player doesn't fold before the river, a hand with at least 14 outs after the flop has a better than 50% chance to catch one of its outs on either the turn or the river. With 20 or more outs, a hand is a better than 2 : 1 favorite to catch at least one out in the two remaining cards.
Estimating probability of drawing outs - The rule of four and twoMany poker players do not have the mathematical ability to calculate odds in the middle of a poker hand. One solution is to just memorize the odds of drawing outs at the river and turn since these odds are needed frequently for making decisions. Another solution some players use is an easily calculated approximation of the probability for drawing outs, commonly referred to as the "Rule of Four and Two". With two cards to come, the percent chance of hitting x outs is about (x × 4)%. This approximation gives roughly accurate probabilities up to about 12 outs after the flop, with an absolute average error of 0.9%, a maximum absolute error of 3%, a relative average error of 3.5% and a maximum relative error of 6.8%. With one card to come, the percent chance of hitting x is about (x × 2)%. This approximation has a constant relative error of an 8% underestimation, which produces a linearly increasing absolute error of about 1% for each 6 outs.
A slightly more complicated, but significantly more accurate approximation of drawing outs after the flop is to use (x × 4)% for up to 9 outs and (x × 3 + 9)% for 10 or more outs. This approximation has a maximum absolute error of less than 1% for 1 to 19 outs and maximum relative error of less than 5% for 2 to 23 outs. A more accurate approximation for the probability of drawing outs after the turn is (x × 2 + (x × 2) ÷ 10)%. This is easily done by first multiplying x by 2, then rounding the result to the nearest multiple of ten and adding the 10's digit to the first result. For example, 12 outs would be 12 × 2 = 24, 24 rounds to 20, so the approximation is 24 + 2 = 26%. This approximation has a maximum absolute error of less than 0.9% for 1 to 19 outs and a maximum relative error of 3.5% for more than 3 outs.
Joined: Jan '11
Location: United States
Age: 44 (M)
Posts: 852
wow, this is one i will have to come back too. Though I have read most of this in super systems and theory of poker. But a very usefull thread. thanks for taking the time.
Joined: Mar '09
Location: Canada
Age: 38 (M)
Posts: 3211
gotta give back to the forum sometime lol not always just a smart ass if someone takes the time to read that didnt already know this then time well spent!
Joined: Jan '11
Location: United Kingdom
Age: 35 (M)
Posts: 704
Posted by arsenej1: gotta give back to the forum sometime lol not always just a smart ass if someone takes the time to read that didnt already know this then time well spent!
i find this very usefull as i havnt read any books on poker. would it be ok for me to use this on my site to help people there to?
Joined: Feb '08
Location: Netherlands
Age: 54 (M)
Posts: 818
Wow Arsenej, this is probably the longest thread i ever saw , must have taken you some time. Sry i diden't read all of it, i allready read about poker probabillities in pokerbooks, but its a good thread and i think the mobsters who haven't read about it will find this very helpfull. Nice1
Joined: Mar '09
Location: Canada
Age: 38 (M)
Posts: 3211
yeah took awhile problem is i hadda cut s**t out and fix everything up now looking thro it there is some stuff that dont line up or symbols got replaced with abuncha ( {A♣, A♥}, this crap i tryed tho lol
Joined: Mar '11
Location: Germany
Age: 57 (M)
Posts: 1242
Thanks for this - a lot of work you did here. Thumbs of for you I guess I have to print it out and read it carefully and see what will help me with my play.
Joined: Oct '09
Location: United Kingdom
Age: 59 (M)
Posts: 667
Thanks for posting this, although I have read through it twice and now am laying down in a darkened room.......................
I have now noticed you have posted the link. I will follow this and hopefully get an idea. Being new to poker this will be helpful if only I can pick some of it up.
Good job. It's something that every poker player should know, but just few really know it. At the micro stakes poker is simply just game of math and good decisions. I'd say you don't need implied odds on lowest limits. Not that you don't need to know how to "calculate" them, but game on micro stakes isn't complicated. You bet strong hands, fold weak. That's all IMO. You don't bluff a lot (I'd say quite never), you don't do spectacular moves. You just need to know when you'rr the best and when you'r behind and you know that all because of the stuff you posted here, so everyone should read it.
Joined: Apr '11
Location: Romania
Age: 37 (M)
Posts: 6669
This is a copy-paste from a online poker book, i'm sure it's also as a book but i personally have read it online. I also think i have it on my pc but i need to do some searching first. This type of probabilities are very usefull in time, when you learn a lot of poker theory and during the game you start to remember all those percents. I could post a link of the great Phil Ivey on Poker After Dark when he knew without doing any math at all the chances for his hand to win on that certain flop and then turn. So i guess if you really want to become a top player you also need to know this probabilities very good. I also know this type of probabilities were used in making bot software by one of their creators. Anyways, very nice post i think many will find this article pretty usefull in their games. Good luck!
Joined: Mar '09
Location: Canada
Age: 38 (M)
Posts: 3211
Posted by dozn01: took 2 hours to copy and paste this looks like a job for dozn01
sure did lol...hadda fix up all the charts and stuff cuz non of it ligned up so had to put it back in origonal place..cut out all proabilities yadayadayada took to long lol coulda just posted origional link at first but.....
Posted by dozn01: took 2 hours to copy and paste this looks like a job for dozn01
sure did lol...hadda fix up all the charts and stuff cuz non of it ligned up so had to put it back in origonal place..cut out all proabilities yadayadayada took to long lol coulda just posted origional link at first but.....
its still hard work mate, what you did, well done on this posts
took some time to round up but seems worth it now! 
, must have taken
i tryed tho lol
You obviciously did your best, and you posted the link, so no prob. they "ll
