Poker Math and Probability

Know your odds

Poker Math is the key element of the game. Such factors as intuition and psychology are not enough. Every decision you make should be mathematically grounded. Knowledge on odds and probability in Texas Hold'em poker is the basis of a successful game.

We'll try not to clutter up the article with a bunch of formulas, but will simply offer a clear table with ready-made calculations:

A table defining the probability of different starting hands in the flop combination

Hits in a flop with speculative hands

SC - suited connector, SC1 - suited connector with a gap, SC2 - suited connector with two gaps, SC3 - suited connector with three gaps; OS – off-suited connector, OC1 – off-suited connector with one gap, OC2 - off-suited connector with two gaps, OC3 - off-suited connector with three gaps; XXs - random suited hand, AXo - random unsuited hand with an ace, XXo - random unsuited hand.

We’ll also cover how to calculate your starting hands’ odds and will teach you to analyze starting hands using simple poker math. Many novices tend to overestimate their starting hands, especially when it comes to suited cards or low pairs. As you can see, suited cards don't always mean you'll collect the flush. Similarly, the pairs make a set only in 12% of cases, and the low pairs are not always worth playing.

Poker Math can be very useful for making the right decision pre-flop. But don't forget that a poker hand consists of five, not two cards. You should also have an idea of what might happen post flop.

If you plan to collect a flush and already hold four suited cards (flush draw), then only nine cards from the deck can help you. These cards are called outs. Each poker deck contains 13 cards of each suit, and thus it's easy to calculate: 13 - 4 = 9. If you hit the flush draw on the flop, then there are 35% odds to collect the fifth best hand to the river. If post turn one card is still missing to the flush, then the probability of completing it to the river is 20%. Knowledge of odds and outs will come in handy later on. These issues will be covered in a separate article of our poker school.

Starting hands' odds against opp's certain hands

Previously we’ve mentioned starting hands' odds as well as the flop', turn' and river' odds in poker. This all is definitely important, as well as to know to measure up your hand against your opponent's. Let's see what starting hands' odds are against other Texas Hold'em poker certain hands.

Pair vs low pair

 vs  (81% vs 19%)

 vs  (82% vs 18%)

Pair vs two high cards

 vs  (53% vs 47%)

 vs  (53% vs 47%)

This is a classic situation that can often be observed when playing Hold'em. As a rule, a pair is a favorite.

Pair vs two low cards

 vs  (88% vs 12%)

 vs  (80% vs 20%)

The pair is usually a 5 to 1 favorite, but the situation may change if the lower cards are suited or connectors.

Pair vs high card and low card

 vs  (70% vs 30%)

 vs  (72% vs 28%)

The pair is a favorite in about 70% of cases. When playing against poker novices, this situation takes place quite often.

Pair vs high card and a card equal in rank to a card in pair

 vs K-spades/6-hearts (66% vs 34%)

 vs (70% vs 30%)

A classic example of this situation is when a strong pair, such as kings, plays against AK. The opponent, in fact, has only three outs on the ace (excluding a possible straight and flush). As you can see, kings have 70% odds of winning.

Pair vs small card and a card equal in rank to a card in pair

 vs  (94% vs 6%)

 vs  (88% vs 12%)

This is an advantageable situation. To win, the opponent should collect at least straight, flush or trips, and his highest card is under domination.

Pair vs suited connectors of a low value

 vs  (78% vs 22%)

 vs  (80% vs 20%)

You must have seen some poker players go all-in with suited connectors. Most often it's because these cards rarely fall under the domination of opponent's cards, and the winning odds here are higher than in the previous example where one of the cards is under domination.

Pair vs suited connectors of high value

 vs  (50,8% vs 49,2%)

 vs  (50,3% vs 49,7%)

If such a situation takes place pre-flop, the winning odds are about the same.

Two high cards vs two low cards

 vs  (69% vs 31%)

 vs  (62% vs 38%)

In this situation, lower cards might provide rather good winning odds, but more often you'll lose.

High card vs two average value cards

 vs  (60% vs 40%)

 vs  (55% vs 45%)

In this case, the hand including the high card is ranked higher than the opp's hand, but the winning odds are not that great.

One high card vs one average value card

 vs  (60% vs 40%)

 vs  (66% vs 34%)

Compared to the previous example, here the winning odds increase by about 5%. It's obvious that the highest card still has the best chances to win.

High card and dominating card vs low card and dominated card

 vs  (75% vs 25%)

 vs  (72% vs 22%)

In this example, the opponent has only three outs (if to ignore possible flush and straight combinations). As for the first case, only 7 will "save" him, in the second one - 3. If your opponent will get a pair with a high card, then your hand will win with an identical pair, provided higher kicker.

High card with a strong kicker vs the same high card with a weak kicker

A-diamonds/K-clubs vs A-hearts/Q-spades (73% vs 27%)

 vs  (73% vs 27%)

This option is quite common in Texas Hold'em poker. It's important to hold a good kicker to increase winning odds in such situations.

All the above calculations give only a rough idea of ​​what to do with this or that card when dealing with your poker opp. However, it might assist our subscribers to more correctly choose their starting hands and to understand what their winning odds are vs those or other cards.