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The theory of probability allows us to estimate how often an event will occur in the game, for example, to determine the chance to collect a certain combination in any of the bidding rounds. The probability is expressed as a percentage from 0% to 100%, a number from 0 to 1 (e.g., 0.33), or as a ratio of favorable to unfavorable outcomes (1 to 2 or 1:2).
1. Probabilities on preflop
Determine how often you will receive preflop pocket aces.
There are 4 aces in a 52-card deck.
The probability that the first of the two pocket cards will be an ace is 4/52. The probability that the second card will also be an ace is 3/51 (3 is how many aces are left in the deck after you get the first ace; 51 is how many cards are left in the deck). To get a pair of aces, both of these events must occur, so multiply 4/52 and 3/51 and get 0.45%. On average, you will get the best starting combination in one of the 222 hands. Similarly, the chance of getting any starting hand can be determined. The probability of obtaining different combinations on preflop is presented in the table below.
Probability of getting a starting hand
| Preflop | Probability |
Pocket Aces   | 0.45% |
Pocket Aces or kings   | 0.90% |
| Any pocket pair | 5.90% |
Ace king suited  | 0.30% |
Ace king offsuited   | 0.90% |
| Ace king any | 1.20% |
| Any two suited cards | 24.00% |
Suited connectors   ,   | 2.17% |
Probability theory also allows us to assess how strong our preflop hand is relative to other players.
- For example, the chances that our opponents at the table have at least one pocket pair older when you have a pocket pair in your hands are collected in the table below.
The probability of a pocket pair seeing a pair older than
| Our hand | 1 player | 2 players | 3 players | 4 players | 5 players | 6 players | 7 players | 8 players |
| Probability of one senior pair (in %) vs. | ||||||||
| 0.49 | 0.98 | 1.47 | 1.96 | 2.44 | 2.93 | 3.42 | 3.91 |
  | 0.98 | 1.95 | 2.92 | 3.88 | 4.84 | 5.79 | 6.73 | 7.66 |
  | 1.47 | 2.92 | 4.36 | 5.77 | 7.17 | 8.56 | 9.92 | 11.27 |
  | 1.96 | 3.89 | 5.78 | 7.64 | 9.46 | 11.24 | 12.99 | 14.7 |
  | 2.45 | 4.84 | 7.18 | 9.46 | 11.68 | 13.84 | 15.93 | 17.95 |
  | 2.94 | 5.8 | 8.57 | 11.25 | 13.84 | 16.34 | 18.73 | 21.01 |
  | 3.43 | 6.74 | 9.94 | 13.01 | 15.95 | 18.74 | 21.38 | 23.87 |
 | 3.92 | 7.69 | 11.3 | 14.73 | 17.99 | 21.04 | 23.89 | 26.51 |
 | 4.41 | 8.62 | 12.63 | 16.42 | 19.96 | 23.24 | 26.23 | 28.92 |
  | 4.9 | 9.56 | 13.95 | 18.06 | 21.86 | 25.32 | 28.41 | 31.09 |
  | 5.39 | 10.48 | 15.26 | 19.67 | 23.7 | 27.29 | 30.4 | 33 |
  | 5.88 | 11.41 | 16.54 | 21.24 | 25.46 | 29.14 | 32.22 | 34.64 |
The chances that the flop, turn or river will not release overcards to our pocket pair are presented below. The probability on the turn is represented as the probability “to the turn” - for 4 cards, and “to the river” - for 5 cards, respectively.
Probability of overcards to reach our pair
Our hand | No overcards on the flop | No overcards on the turn | No overcards on the river |
(probability in %) | |||
| 77.45 | 70.86 | 64.7 |
| 58.57 | 48.6 | 40.15 |
| 43.04 | 32.05 | 23.69 |
| 30.53 | 20.14 | 13.13 |
| 20.71 | 11.9 | 6.73 |
| 13.27 | 6.49 | 3.1 |
| 7.86 | 3.18 | 1.24 |
| 4.16 | 1.33 | 0.4 |
| 1.86 | 0.43 | 0.09 |
| 0.61 | 0.09 | 0.01 |
| 0.1 | 0.01 | <0.01 |
Probability of coming under direct dominate with AX hands (or AK to AK) against a certain number of players after us
Our hand | 1 player | 2 players | 3 players | 4 players | 5 players | 6 players | 7 players | 8 players |
| Probability of direct dominate | ||||||||
| 0.24 | 0.49 | 0.73 | 0.98 | 1.22 | 1.46 | 1.7 | 1.94 |
| 1.22 | 2.43 | 3.63 | 4.81 | 5.97 | 7.13 | 8.26 | 9.39 |
| 2.2 | 4.36 | 6.47 | 8.63 | 10.55 | 12.52 | 14.45 | 16.33 |
| 3.18 | 6.27 | 9.25 | 12.14 | 14.94 | 17.65 | 20.27 | 22.81 |
| 4.16 | 8.15 | 11.98 | 15.64 | 19.15 | 22.52 | 25.75 | 28.84 |
| 5.14 | 10.02 | 14.65 | 19.04 | 23.2 | 27.15 | 30.9 | 34.45 |
| 6.12 | 11.87 | 17.27 | 22.33 | 27.09 | 31.55 | 35.74 | 39.67 |
| 7.1 | 13.7 | 19.83 | 25.52 | 30.61 | 35.73 | 40.29 | 44.53 |
| 8.08 | 15.51 | 22.34 | 28.62 | 34.38 | 39.69 | 44.56 | 49.04 |
| 9.06 | 17.3 | 24.79 | 31.61 | 37.81 | 43.44 | 48.57 | 53.23 |
| 10.04 | 19.07 | 27.2 | 34.51 | 41.08 | 47.00 | 52.32 | 57.11 |
| 11.02 | 20.83 | 29.55 | 37.31 | 44.22 | 50.37 | 55.84 | 60.71 |
These figures are indicative of the opposition of early and late positions and explain why the secret game in this case is mathematically justified.
2. Probabilities on post-flop
Similarly, the chances of assembling combinations of different strengths on a flop can be determined.
Probability of assembling a combination on a flop
Flop | Probability |
Pair | 32.4% |
Two pair (from unpaired cards) | 2% |
Set | 11.80% |
Straight | 1.3% |
Straight draw | 10.50% |
Flush | 0.84% |
Flush draw with two suited pocket cards | 10.9% |
Full-house with pocket pair | 0.70% |
Caret with pocket pair | 0.25% |
On the flop, you also need to know what the chances are that you or your villain will improve the hand.
- For example: On the preflop, the player has a one-matted hand, and on the flop, two more cards of the same suit appear
To collect the flush, he needs one of the remaining nine cards of this suit on the turn or river. In this case, the player has nine outs to collect probably the best hand (an “outs” in poker terminology is any desired card that will strengthen the hand and potentially lead it to victory). In percentage terms, the chance to collect flush on the turn is 19.1%, on the river (if the turn did not help) - 19.6%. The probability of collecting a flush on a turn or river is 35%. The chances of increasing on the postflop, depending on the number of outs, are shown in the table.
The likelihood of getting the necessary outs on the following streets of betting
Outs | Chance of gain | Probability of gain | Probability of gain |
| 20 | 42.6% | 43.5% | 67.5% |
19 | 40.4% | 41.3% | 65.0% |
18 | 38.3% | 39.1% | 62.4% |
17 | 36.2% | 37.0% | 59.8% |
16 | 34.0% | 34.8% | 57.0% |
15 | 31.9% | 32.6% | 54.1% |
14 | 29.8% | 30.4% | 51.2% |
13 | 27.7% | 28.3% | 48.1% |
12 | 25.5% | 26.1% | 45.0% |
11 | 23.4% | 23.9% | 41.7% |
10 | 21.3% | 21.7% | 38.4% |
9 | 19.1% | 19.6% | 35.0% |
8 | 17.0% | 17.4% | 31.5% |
7 | 14.9% | 15.2% | 27.8% |
6 | 12.8% | 13.0% | 24.1% |
5 | 10.6% | 10.9% | 20.3% |
4 | 8.5% | 8.7% | 16.5% |
3 | 6.4% | 6.5% | 12.5% |
2 | 4.3% | 4.3% | 8.4% |
1 | 2.1% | 2.2% | 4.3% |
Examples of calculation per one street:
- Flush draw (9 outs): 9 * 2 = 18%
- Straight draw (8 outs): 8 * 2 = 16%
- Two pair and you need to build a full-house (4 outs): 4 * 2 = 8%
Multiply your outs by 4 when your villain goes all-in on the flop. 9 outs with flush draw give you 36%, which is very close to the real 35% Chances to increase on the turn and river, being on the flop with combinations of different strengths, are presented in the table below.
Probability to improve the combination
Situation | Probability for | Probability of |
Set to quads | 2.13% | 4.26% |
Pocket pair to set | 4.26% | 8.42% |
Pair to two pair | 6.38% | 12.49% |
Gutshot | 8.51% | 16.47% |
One pair to two pair or thrips | 10.64% | 20.35% |
Two overcards to pair | 12.77% | 24.14% |
Set to full house or quads | 14.89% | 27.84% |
Straight draw to the street | 17.02% | 31.45% |
Flush draw to flush | 19.15% | 34.97% |
Gutshot and two overcards to a straight or pair | 21.23% | 38.39% |
Straight draw and one overcard to straight draw or pair | 23.40% | 41.72% |
Flush draw and one overcard to flash or pair | 25.53% | 44.96% |
Flush draw and gutshot to flush or straight | 27.66% | 48.10% |
Flush draw and two overcards to flash or pair | 29.79% | 51.16% |
Straight draw and flush draw to straight or flush | 31.91% | 54.12% |
Straight draw and flush draw with two overcards | 44.68% | 69.94% |
3. Summary
Probability theory helps us estimate how profitable an action will be. Knowing the poker probabilities allows you to adjust the strategy during the game, makes the expectations of the results reasonable and helps to maintain emotional stability in order to continue playing your best poker.
Further articles on the basics of poker mathematics: Thinking in ranges is a key skill of successful poker players, pot odds in poker or how to calculate the profitability of a decision, What is equity in poker, and why is it so important to understand this?, fold equity in poker and the mathematics of bluff, The principle of narrowing the range is the basis of the strategy of playing poker.
