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Overview: Poker math

Poker is a game that demands several skills be used at once, and the best players also use some basic mathematics to help them make the correct decisions. It is not necessary to have a really complex mathematical mind to play poker - so don't be discouraged - but there are a few elements to the game that all players should learn.

We have already seen how the relative strength of a poker hand can increase or decrease as flop, turn and river is dealt. For example  is a big favourite against  pre-flop, but becomes a huge underdog if the flop comes .

If you have a hand that is probably behind, but has the potential to improve to a winner, you need to decide whether it is worth continuing with it through the various streets, and how much you are prepared to pay to do so.

This article explains the calculations required to make the right decision about "drawing hands", ie, hands that will need to connect with later community cards to win. 

The first step is to identify the cards that will improve your hand (known as "outs"). Once you have managed that, you can move on to the calculating how they might help you.

CALCULATING OUTS

"Outs" are the cards left in the deck that improve your hand and will help you win the pot at showdown. The best way to demonstrate what we mean by outs is to look at a few common examples:

Example with a flush draw:
You are holding  and the flop is: . If another heart appears on the turn or river, you make a flush, and unless another player has a full house or better, you will win the hand. (The board isn't paired, so none of our opponents can have a full house yet.)

There are 13 cards of each suit in the deck. You hold two of them, and another two are on the board. Four of the 13 hearts have therefore already been dealt, meaning that there are still nine hearts left in the deck.

This means you have nine cards that can improve your hand to a probably winner. You have nine outs.

Example with a straight draw:
You have  and the flop is . Now any ace or nine will complete your straight. There are four aces and four nines in the deck, so you have eight outs.

If one card is missing to complete a straight, you have four outs. For example, if your hole cards were  and the flop was , your outs would be .

 

Example with a straight draw and overcards:
You have , and the board is . One of the four queens in the deck will make you a straight. If your opponent has a middle pocket pair, e.g. , then you have additional outs, as any king or any jack would give you a higher pair. 

In this case, the number of your outs would increase to ten (four queens, three kings, and three jacks).

 

Example with a set against a flush draw:
If you hold  and hit a set on a board showing , you have a pretty strong hand. But it is not definitely a winner and could already be behind if any of your opponents has two spades in his hand.

However, you still have the chance here of improving your hand even further. There are seven cards that could make you a full house or better (a seven, three remaining twos and three remaining jacks), or the turn and river could be the same rank, which would also give you a full house.

 

Example with a straight and a flush draw:
You hold  and the board is . You have both an open-ended straight draw and a flush draw. This means you have nine outs to make the flush and eight outs to make the straight. At the same time, you have to consider that two cards are counted twice (in this case the  and the ), which have to be subtracted. Therefore you have a total of 15 outs here.

HIDDEN OUTS

Although the term "out" typically refers to a card that improves your hand, there are also sometimes "hidden outs", which help you because they reduce the value of your opponent's hand.

 

Example of hidden outs:
You hold  and your opponent has . The board is . Not only would the three kings and the three aces give you a higher two pair than your opponent, but any six or five would help as well.

This is because with a five or six, the board contains two pairs that are both higher than your opponent's pocket threes, meaning that the fifth card, the kicker, would decide the outcome of the hand. Your ace is the best possible kicker. 

In this instance, you have 12 outs, six of which are hidden.

 

DISCOUNTED OUTS

Advanced players don't only calculate their own outs when on a draw. They also ask themselves what hand their opponent has, and whether one of the cards they hope to appear might also give the other player an even better hand.

Cards like this are known as "discounted outs".

The straight draw example again:
You have  and the flop is . You have calculated eight outs so far (four aces and four nines).An example of when to discount outs during a Texas Hold'em poker hand.

But how will your outs change if one your opponents has two hearts e.g.  and is therefore drawing to a flush?

In this example, two of your outs, i.e.  and , would give your opponent a better hand - even if you hit your straight.

This means you have to discount both cards from your outs. You would now only have six outs, which significantly reduces your chances of winning the hand.

In general you should take a pessimistic approach when it comes to discounting outs, as it is better to discount one out too many than one too few!

FURTHER READING

The subject of poker mathematics can get very complicated. If you are of a mathematical mind, you might want to read more on the subject in an advanced continuation of this outs and odds article. However the basic course continues with a concenient short-cut known as the "Rule of Two and Four".

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