Learn the Poker Probabilities at the Table
In most cases, the probabilities and poker hand odds are approximations due to rounding but follow a direct poker mathematic formula.We use a basic approach.
Determine number of outcomes (divided by) possible outcomes.
To pick a 2 card combination from a standard 52 card pack. You would have 52 outcomes for 1st card & 51 options for 2nd card (ignoring order). 52 × 51 ÷ 2 = 1,326
OK here's a little trick. So you have 4 cards & want 2 combination.You have for example only 4 aces in the pack & you are looking for a 2 card combination to make a pair of aces.
Simple formula 4x3 ÷ 1x2 = 12 ÷ 2 = 6
If it was a 3 card formula 4x3x2 ÷ 1x2x3 = 24 ÷ 6 = 4 ( but lets stick to 2 cards for illustration)
Putting it together 1326 outcomes ÷ 6 combinations = 221 chance of being deal 2 aces
Starting hands
Simple poker mathematics can be used to calculated a holdem starting hand or hole cards .Taking the 1st card will be any one of 52 & the 2nd will be any one of the 51 remaining cards. This gives
52 × 51 ÷ 2 = 1,326 (possible starting hand combinations. Divide by 2 because the order is irrelevant.)
The 1,326 starting hands can be reduced when determining the probability of starting hands in Holdem as suits have no value before the flop. The only factors determining the strength of a starting hand are the card rank & if they are suited cards.
. Of the 1,326 combinations, there are 169 distinct starting hands grouped into parts:
13 pocket pairs (i.e., 2,3,.......QKA) × 6 combinations = 78 suited hands instead of ÷ 2, we will add the 78 unsuited hands. To put this together.
13 + 78 + 78 = 169 (distinct starting hands)
The relative probability of being dealt a hand of each groups different. The following shows the probabilities and odds of being dealt each type of starting hand.
Pocket pair ---- 13 hands x (suit permutation) 6 = 78 ÷ 1326 outcomes = 0.0588 or 16 : 1 Odds
Suited cards---- 13 hands x (suit permutation) 4 = 312 ÷ 1326 outcomes = 0.2353 or 3.35 : 1 Odds
Unsuited cards-- 13 hands x (suit permutation) 12 = 936 ÷ 1326 outcomes = 0.7059 or 0.417 : 1 Odds
Here are the probabilities and poker hand odds of being dealt various other types of starting hands.
AK's (or any specific suited cards), 0.00302 331 : 1 Odds
AA (or any specific pair), 0.00453 220 : 1 Odds
AK's, KQ's, QJ's, or JT's, 0.0121 81.9 : 1 Odds
AK (or any specific non-pair), 0.0121 81.9 : 1 Odds
AA, KK, or QQ 0.0136, 72.7 : 1 Odds
Suited cards, J or better, 0.0181 54.3 : 1 Odds
AA, KK, QQ, JJ, or TT, 0.0226 43.2 : 1 Odds
Suited cards, T or better, 0.0302 32.2 : 1 Odds
Suited connectors, 0.0392 24.5 : 1 Odds
Connected cards, T or better, 0.0483 19.7 : 1 Odds
Any 2 cards with rank at least Q, 0.0498 19.1 : 1 Odds
Any 2 cards with rank at least J, 0.0905 10.1 : 1 Odds
Any 2 cards with rank at least T, 0.143 5.98 : 1 Odds
Connected cards (cards of consecutive rank) 0.157 5.38 : 1 Odds
Any 2 cards with rank at least 9, 0.208 3.81 : 1 Odds
Not connected nor suited, at least one 2-9, 0.534 0.873 : 1 Odds
Starting hands heads up
For any given starting hand, there are50 × 49 ÷ 2 = 1,225 hands that an opponent can have before the flop. After flop, discount the 3 flop cards to determine the opponents possibilities 47 × 46 ÷ 2 = 1,081
Head-to-head starting hand match ups
When comparing two starting hands, the head-to-head probability shows the chance of one hand beating the other after all the cards are out. Head-to-head probabilities vary a little for each particular distinct starting hand match up, but here is a rough estimation of the poker hand odds.
Pair vs. 2 under cards, 0.83 4.9 : 1 Odds
Pair vs. lower pair, 0.82 4.5 : 1 Odds
Pair vs. 1 over card, 1 under card0.71 2.5 : 1 Odds
2 over cards vs. 2 under cards, 0.63 1.7 : 1 Odds
Pair vs. 2 over cards, 0.55 1.2 : 1 Odds
These odds rough calculations by averaging all of the hand match ups in each category. The actual head-to-head probabilities are dependant on a multitude of factors..........
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.
Dominated hands
When evaluating the poker hand odds before the flop, it's useful to know how likely the hand is dominated. A dominated and is extremely unlikely to win. Often the dominated hand has only a single card rank that can improve the dominated hand to beat the dominant hand (not counting straights and flushes). For example, AJ is dominated by AQ—both hands share the Ace and the queen kicker is beating the jack kicker. Barring a straight or flush, the AJ requires a Jack on the board to improve against the AQ (and will lose if a queen also lands).Pocket pairs are dominated by higher pocket pairs.
The flop
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 7s, any hand holding the fourth 7has the nuts. Conversely, the flop can undermine the perceived strength of any hand—a player holding A clubs, A spades would not be happy to see 9 10 Jack of hearts on the flop because of the straight and flush.
The following are some general probabilities about what can occur on the board. These assume a "random" starting hand for the player.
Three or more of same suit 18.3 : 1 to make of flop, 6.40 : 1 to make on turn & 3.24 : 1 river
Four or more of same suit to make on turn 93.7 : 1, river 28.5 : 1
Rainbow flop (all different suits) 1.51 : 1 to make of flop , 8.48 : 1 river
Three cards of consecutive rank (but not four consecutive) 27.8 : 1 flop, 7.46 : 1 turn, 2.99 : 1 river
Four cards to a straight (but not five) 24.8 : 1 to make on turn, 4.27 : 1 river
Three or more cards of consecutive rank and same suit 459 : 1 flop, 14 : 1 turn, 45.0 : 1 river
Three of a kind (but not a full house or four of a kind) 424 : 1 flop, 106 : 1 turn, 46 : 1 river
A pair (but not two pair or three or four of a kind) 4.90 : 1 flop, 2.29 : 1 turn, 1.36 : 1 river
Two pair (but not a full house) 95.4 : 1 turn, 20.2 : 1 river
Note that more than 60% of flops have at least two suited so some fish who has no knowledge of poker mathematics could be chasing their flush opportunities.
Probability that no over cards will come on the flop, turn and river .Holding pocket pair No over card on flop No over card by turn No over card by river
KK 0.29 : 1 flop 0.41 : 1 turn 0.55 : 1 river
QQ 0.71 : 1 flop 1.06 : 1 turn 1.49 : 1 river
JJ 1.32 : 1 flop 2.12 : 1 turn 3.22 : 1 river
TT 2.28 : 1 flop 3.97 : 1 turn 6.61 : 1 river
99 3.83 : 1 flop 7.40 : 1 turn 13.87 : 1 river
88 6.54 : 1 flop 14.40 : 1 turn 31.21 : 1 river
77 11.73 : 1 flop 30.48 : 1 turn 79.46 : 1 river
66 23.02 : 1 flop 74.26 : 1 turn 246.29 : 1 river
55 52.85 : 1 flop 229.07 : 1 turn 1,057.32 : 1 river
44 162.33 : 1 flop 1,095.67 : 1 turn 8,406.78 : 1 river
33 979 : 1 flop 15,352.33 : 1 turn 353,125.67 : 1 river
Using simple Poker Mathematic tables like these can give you a slight edge when determining your betting strategy and working out the poker hand odds.
Using Outs to calculate Odds of winning
From the flop onwards drawing probabilities are calculated as outs. All situations which have the same number of outs have the same probability of winning. E.g., an inside straight draw 8-9-10-Q needs the J to complete the straight, and a full house draw QQ-KK are equivalent. Each have 4 out to make the desired hand, i.e., any 1 of 4 J's in the pack for the straight. either of the 2 Q's or K's to complete the full house.
Using poker mathematics to calculate.
At flop = 47 unseen cards so the probability = (outs ÷ 47).
At turn = 46 unseen cards so the probability - (outs ÷ 46).
Most poker players don't have the mathematical ability to calculate the poker hand odds mid play, so memorizing a simple list of potential outs helps immensely in their poker decisions. 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.
| Outs |
Likely Drawing Hands |
Odds on Turn |
Odds on River |
Either Turn or River |
| 1 |
Inside straight flush; Four of a kind turn |
46.0 : 1 |
45.0 : 1 |
22.5 : 1 |
| 2 |
Open-ended straight flush; Three of a kind turn |
22.5 : 1 |
22.0 : 1 |
10.9 : 1 |
| 3 |
High pair |
14.7 : 1 |
14.3 : 1 |
7.01 : 1 |
| 4 |
Inside straight; Full house |
10.8 : 1 |
10.5 : 1 |
5.07 : 1 |
| 5 |
Three of a kind or two pair |
8.40 : 1 |
8.20 : 1 |
3.91 : 1 |
| 6 |
Either pair Full house or four of a kind |
6.83 : 1 |
6.67 : 1 |
3.14 : 1 |
| 7 |
Inside straight or high pair |
5.71 : 1 |
5.57 : 1 |
2.59 : 1 |
| 8 |
Open-ended straight |
4.88 : 1 |
4.75 : 1 |
2.18 : 1 |
| 9 |
Flush |
4.22 : 1 |
4.11 : 1 |
1.86 : 1 |
| 10 |
Inside straight or pair |
3.70 : 1 |
3.60 : 1 |
1.60 : 1 |
| 11 |
Open-ended straight or high pair |
3.27 : 1 |
3.18 : 1 |
1.40 : 1 |
| 12 |
Inside straight or flush; Flush or high pair |
2.92 : 1 |
2.83 : 1 |
1.22 : 1 |
| 13 |
|
2.62 : 1 |
2.54 : 1 |
1.08 : 1 |
| 14 |
Open-ended straight or pair |
2.36 : 1 |
2.29 : 1 |
0.955 : 1 |
| 15 |
Inside straight, flush or top pair. Open-ended straight or flush; Flush or pair |
2.13 : 1 |
2.07 : 1 |
0.848 : 1 |
| 16 |
|
1.94 : 1 |
1.88 : 1 |
0.755 : 1 |
| 17 |
|
1.76 : 1 |
1.71 : 1 |
0.673 : 1 |
| 18 |
Open-ended straight, flush or high pair Inside straight or flush or pair; |
1.61 : 1 |
1.56 : 1 |
0.601 : 1 |
| 19 |
|
1.47 : 1 |
1.42 : 1 |
0.538 : 1 |
| 20 |
|
1.35 : 1 |
1.30 : 1 |
0.481 : 1 |
| 21 |
Open-ended straight, flush or pair |
1.24 : 1 |
1.19 : 1 |
0.430 : 1 |
" Using Poker Mathematics will dramatically improve your game of Texas Holdem." Work on understanding the poker hand odds to dominate your rivals
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