Calculating poker probabilities


So, at the very beginning of your journey in the topic of statistics and probabilities in poker, we will figure out what kind of cards and with what probability the players of Texas poker receive at the very beginning of the hand. According to the rules of the game in hold’em, at the start of a new hand, all poker players receive 2 pocket cards. At the same time, there are 52 cards in the deck, which means that the total number of different combinations of two cards will be 1326 (52×51 / 2 = 1326). In this case, the calculation of the probability in poker hold’em to receive a specific pair of pocket cards will look like this: 4×3 / 2 = 6. Thus, making a simple calculation of the probability in poker,we will see that a specific pair of hole cards will come to us in each of the 221 hands (6/1326 = 1: 221). Since poker 52 cards may collect only 13 pocket pairs, any pocket pair will be your starting hand at every 17th hand (221/13).

When trying to calculate the probability in poker for starting hands, you must also consider the suit of the cards received. So, any 2 unpaired cards can be dealt to the players at the start of the distribution in 16 different variants (suited and unsuited). In this case, any two suited cards will be collected in 23.5% of cases (12/51), and cards of a particular suit in 5.9%. Calculating hands in poker is not limited to calculating preflop statistics – there is a section ahead of you on the odds of different starting hands to be strengthened on different streets of the hand. At the same time, online players should take into account that suited starting cards will always add an additional 1-2% to the probability of winning the game .

Poker hand odds

At the flop stage, we already have 3 out of five cards on the board, which means that this very moment in the game is the most important from the point of view of the theory of probability in poker. At this stage, most of the distribution remains completed and the outcome of the entire distribution often depends on the decisions made here. And right now, the most important thing for us is to correctly calculate the probabilities of combinations in poker, as well as to find out the objective strength of our hand, which will allow us to get the long-awaited victory. In online money games , players most often do not reach the showdown, however, for convenience, we will consider the probabilities under the condition that the distribution goes to the very end, and the poker players open their cards at the last stage.

So, to start with, the probability of getting a combination in poker with unpaired cards on the flop will be as follows:

  • A pair on the flop – 26.95%
  • Two pairs, of which one pair + one card of the second pair – table cards – 2.02%.
  • Two pair, where both of the player’s cards coincide with the two cards of the board – 2%
  • Trips of one pocket card and two table cards – 1.35%
  • Full House – 0.09%
  • Caret – 0.01%

Thus, summing up all the received odds of combinations in poker on the flop with an unpaired pocket, we get a probability of a poker combination of 32.4%, which means that even with unpaired hand cards at the start of the hand in Texas Hold’em, the player will be able to improve to the flop almost in a third of cases.

The probabilities of combinations in Texas Hold’em for players with pocket pair will look like this:

  • Two pair: one on the table, the other from hand – 16.6%
  • Set – 10.77%
  • Full House (a pair and one trips card that matches the pocket) – 0.74%
  • Full House (three of a kind and pocket pair) – 0.25%
  • Caret – 0.25%

It is also worth noting that two suited cards in the starting hand give the player a 0.84% ​​chance of getting a flush on the flop, but such a poker player will manage to make a flush draw 10.94% of the time. Offsuit cards in the hand will only make a flush draw in 2.25% of the outcomes.

Hand connectors will flop a straight or straight draw with the following odds:

  • Connectors in series – 1.31% for straight and 10.45% for straight draw
  • Single Hole Connectors – 0.98% for straight and 8.08% for straight draw
  • With two holes – 0.65% and 5.22%, respectively
  • With three holes – 0.33% and 2.61%

It is also worth remembering that with 56 connectors, you can line up straights in both directions (23456 and 56789), while the 789TJ hand, having collected the best straight using the community cards of the table, will beat your hand. With a hand like AK or A2, you can only make one straight each, and such connectors will have the same chances of making a combination in poker as connectors with three holes.

The turn rarely plays a decisive role in the game , and therefore we will not stop at this stage of trading now, but the river, as the culminating moment of the distribution, deserves our close attention, because it is after this street that you will have to show your cards. What is the chance of poker combinations falling on the last street, provided that the player does not have a single combo in his hands yet?

If your starting hand contains two unpaired cards:

  • Couple will gather at 46%
  • Two pairs – 22%
  • Triplet – 4.5%
  • Full House – 2.2%
  • Caret – 0.1%

If the starting hand has a pocket pair:

  • Two pairs – 40%
  • Set – 12%
  • Full House – 8.5%
  • Caret – 0.84%
  • Flush – the chance of a combination in poker with two suited cards on hand will be 6.6%. A royal flush without two cards on the flop will be 0.05%, and a straight flush with connectors of the same suit will be 0.2%.

An ordinary straight – the probability of a combination in poker with connectors on hand will be considered separately for all types of pockets:

  • Holeless connectors – 9.1%
  • With one hole – 7.8%
  • With two holes – 6.5%
  • With three – 5.1%.

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