# The mathematics of poker. Probabilities

The mathematics of poker is an integral part of the game. Intuition and knowledge of psychology are not enough. Every decision you make must have a mathematical justification. Knowing the odds and probabilities in Texas Hold’em poker is the basis for a successful game.

The math of poker can be very helpful in making the right decision in the preflop. But remember that a hand in poker consists of five cards, not two cards. It is helpful to have an idea of how things can develop after the flop.

If you want to make a flush, and you already have four cards of the same suit (a flush draw), only nine cards from the deck can help you. These cards are called outs in poker. Keep in mind that there are only 13 cards of each suit in the deck, so it’s easy to calculate 13 – 4 = 9. If you have a flush draw on the flop, you have a 35% chance of making the fifth strongest hand on the river. If you’re still one card short of a flush after the turn, you have a 20% chance of completing it on the river. Information on odds and outs will come in handy next. That’s why we suggest discussing pot odds and the need for certain bets with our free ProMTT 3.0 course trainer.

## The odds of starting hands against specific hands of your opponent

So far we’ve talked about the odds of your starting hands, as well as the odds on the flop, turn, and river in poker. These are all obviously important, but it’s also important to have an idea of what cards your opponent might have. Let’s take a look at what the odds are of starting hands against other certain hands in Texas Hold’em poker.

Pair vs. small pair
The math in poker hold’em simple math in poker vs (81% vs. 19%)
Simple poker math vs. easy poker math(82% vs. 18%)

Pair vs. two high cards
vs. (53% vs. 47%)
Versus (53% vs. 47%)

This is a classic situation often seen when playing Hold’em. As a rule, a pair is the favorite.

Pair vs. two junior cards
Versus (88% vs. 12%)
Versus (80% vs. 20%).

Pair is usually a 5-to-1 favorite in this situation, but that can change if the junior cards are the same suit or are connectors.

Pair vs. a high card and a low card
Opposite (70% vs. 30%)
(72% vs. 28%)

Pair is the favorite approximately 70% of the time. When playing against beginners in poker, this situation occurs quite often.

Pair vs. a high card and a card of equal value to a card in a pair
Against (66% vs. 34%)
Versus (70% vs. 30%).

A classic example of this situation is when a strong pair, such as kings, plays against AK. The opponent in fact only has three outs for an ace (excluding a possible straight and flush). From this we can see that kings have a 70% chance of winning.

Pair vs. a small card and a card of equal value to the card in the pair
Versus (94% vs. 6%)
Versus (88% vs. 12%).

This is a plus situation. To win, your opponent must collect at least a straight, flush, or triplet, and his high card is under domination.

Pair vs. low-value singles connectors
Against (78% vs. 22%)
Versus (80% vs. 20%)

You’ve probably seen some poker players go all-in with single-handed connectors. More often than not, this is because these cards will rarely get dominated by their opponent’s cards, and the odds of winning are higher than in the previous example, where one of the cards is dominated.

Pair vs. high-denomination one-handed connectors
Against (50.8% vs. 49.2%)
Versus (50.3% vs. 49.7%)

If this situation occurs on the preflop, the odds of winning are about the same.

Two high cards vs. two low cards
Against (69% vs. 31%)
vs (62% vs 38%).

In this situation, the low cards have a pretty good winning percentage, but you’ll still lose more often than not.

Senior card vs. two medium cards
Versus (60% vs. 40%)
Versus (55% vs. 45%).

In this case, the hand that includes the high card is valued higher than your opponent’s hand, but the odds of winning are not as great.

One high card vs. one medium card
Versus (60% vs. 40%)
Versus (66% vs. 34%).

Compared to the previous example, the percentage to win here goes up about 5%. Obviously, the high card still has the best chance of winning.

Senior card and dominant card vs. junior card and dominant card
Versus (75% vs. 25%)
vs (72% vs 22%).

In this example, the opponent has only three outs (if you neglect the possible flush and straight combinations). In the first case, only a 7 would help him, and in the second, a 3. If your opponent makes a pair with a high card, your hand will win with the same pair but a high kicker.

A high card with a strong kicker vs. the same high card with a weak kicker
Versus (73% vs. 27%)
Versus (73% vs. 27%)

This option is also quite common in Texas Hold’em poker. It is very important to have a good kicker in poker for high odds of winning in these situations.