The exciting and challenging game of Texas Hold’Em Poker is gaining more followers daily-both young and old-who enjoy competing, bluffing, and most of all, winning. In this guidebook for both novice and experienced poker players, an avid poker player combines his knowledge of the game with his expertise regarding stats and probabilities to provide all possible outcomes of the cards, ultimately helping poker players around the world improve their chances of winning.

Sam Habash has been playing poker since he was a young boy, but in the last ten years, has focused mainly on improving his Texas Hold’Em game. He draws on his ability to gather and analyze statistics to provide poker players with probabilities and payback information. While offering little-known statistics like the likelihood of having an ace or king show up in a hand (62%) or the odds of receiving a pair (6%), Habash leads players through many different scenarios like the chances of receiving a second pair, flush draw, or complete straight.

The expert advice offered in this unique reference manual will help any poker player gain the extra advantage to come out ahead in their next game of Texas Hold’Em!

TO THE RIVER

Odds and Probabilities in Texas Hold'em PokerBy Sam Habash

iUniverse, Inc.

Copyright © 2011 Sam Habash
All right reserved.
ISBN: 978-1-4502-8435-6

Contents

Introduction......................................viiChapter One Getting Started......................1Chapter Two Player Cards.........................7Chapter Three One Pair...........................13Chapter Four Two Pair............................17Chapter Five Three of a Kind.....................21Chapter Six Straights............................25Chapter Seven Flushes............................29Chapter Eight Full House.........................33Chapter Nine Four of a Kind......................37Chapter Ten Straight Flush.......................39Chapter Eleven Examples..........................41Summary...........................................71

Chapter One

Getting Started

The game of Texas Hold'em poker is played in two major varieties, limit or no-limit. In limit poker, the casino or a house rule determines the bet limits (in terms of size) and number of allowed raises. No-limit poker is obviously very simple—there are no limits on any of the bets.

The game of Texas Hold'em is typically played with between six and ten players. Each player is dealt two down cards. To induce the betting, the two players to the left of the dealer are required to blind bet, and these are referred to as the small and big blinds. In addition to the two down cards (also called player cards), the dealer reveals five community cards, which are used by all players. These community cards will be opened at three different stages in order to induce betting.

Each player looks to make the best poker hand they can using any five of the seven total cards.

The Betting Process

There are four rounds of betting total:

First bet Made after the dealing of the player cards

? The small and big blinds are bet automatically, before the dealing of the cards ? Betting here is induced by the blinds ? Players must match or raise the big blind

Second bet Made after revealing of the flop (which consists of three community cards that are opened together)

? Bet sizes can vary depending on the game but can be from a minimum of the size of big blind to a maximum of the declared limit, or no limit

Third bet Made after revealing the turn card (the fourth card of five community cards)

Fourth bet Made after revealing the river card (the fifth card of the community cards)

Please note that any player can bet at any time, but betting is typically done in a sequential order starting to the left of the dealer.

Dependent Probabilities

In order to clarify some of the probability calculations in the book, I will briefly attempt to explain the theory of dependent probabilities and difference between those and non-dependent probabilities.

A player who is looking to flop a set (the common term for three of a kind) can do so if he has a pair in his hand that matches a card that comes on the flop. The probability of this is as follows:

Probability of being dealt a pocket pair 6% Probability of matching a set on the flop 12.2% (2/50 × 3 = 12.2%) (2 out of 50 cards, and given three opportunities)

Total of above probability 0.73% (less than one percent)

As you can see, the probability of starting with a pocket pair and the probability of matching the set on the flop had to be multiplied by each other to get the final probability. First you have to get a pair (probability 6/100) then match a card on the flop (12/100), with the overall probability being about .007 (seven hands out of one thousand). This is the concept of dependent probability. The first event has to happen in order for the second event to occur.

But since we all get to see our player cards, if you have already been lucky and have been dealt a pair in your hand, the probability of getting a set is on the flop will be 12.2% because you've already got the pocket pair. Therefore, the probability of matching a set over the entire course of the hand is as follows:

On the flop = 12.2% On the turn = 4.2% On the river = 4.3% Total 20.6%

So the probability of matching a pocket pair and getting a set is about 20%. But if you don't match it on the flop or the turn, the probability of matching it on the river is about 4%.

The probabilities are not dependent on each other since you got to see the card at each stage.

Payback Numbers

The payback number is the ratio of pot size to bet size. You will see this number throughout this book. The question the payback number helps you answer is, should you make that call on the turn or river card? The answer depends on the probability of your winning the hand.

Simply put, say you are on a draw and that your chances of winning the hand are 25%. In this case, the payback number is four to one, which is (100 / 25) and is also written as 4:1. Since your chances of winning are only 25%, you should be getting 4:1 odds in order to make your gamble mathematically worthwhile.

So just to clarify, if you are calling a $10 bet here and your chases of winning is 25%, the pot has to be at least 4 times the size of the bet.

As another example, if you believe that you have to match a set on the river (meaning you have a pocket pair in your hand) and you need to hit a set on the river to win.

The probability is 4.3% so the payback number is (100 / 4.3) = 23.2

so the pot-size-to-bet ratio has to be minimum of 23 to 1.

This means that if your competitor's bet was $10, the pot size has to be at least $230 to make this call mathematically feasible for you. So if the pot was much bigger than that, you should make the call, but if it was smaller than $230, then mathematically it is not wise.

Here is one more example: You are on a flush draw after the flop. The probability of completing the flush is 39%. Therefore, the payback number is approximately 2.5:1. So the pot size has to be two-and-one-half times the size of the bet to make calling mathematically feasible.

However, we all know that sometimes we are pot committed, depending on total stack situation and the size of the blinds. Remember, this book does not tell you what to do; it only provides you with the mathematical probabilities. The rest is up to you.

Chapter Two

Player Cards

I will not spend too much time on this subject, since most of the bidding in Texas Hold'em is done after the players review their cards. However, it is interesting to know certain probabilities, such as that of being dealt an Ace/King starting hand, or a pair.

Probability of having a pair 5.9%

Probability of having a combination of high cards 7.2% (any combination of Aces and face cards)

Total 13.1%

So as can be seen from above, the probability of getting a pair or any combination of high cards is around 13%. This means that if you are a player who plays only starting hands that consist of a combination of high cards or a pair, you will probably play around thirteen hands out of each one hundred dealt to you.

The probability of finding an Ace in the deck of cards is 1 in 52, or 7.7%, and that same probability applies to any specific card. So what is the probability of the following card combinations being dealt to you? Let's look at pairs, for example:

A. Pair of Aces 0.45%

For every two hundred hands played, you will get a pair of Aces once.

B. Any high pair 1.8%

(Aces, Kings, Queens, or Jacks)

C. Any pair higher than nines 2.7%

You will get a pair of Tens or higher in one out of every thirty-seven hands.

D. Any pair 5.9%

Now look at the probabilities of high card combinations:

A. Ace/King 1.2%

(also called "big slick") As can be seen, getting Ace/King is about twice as likely as getting a pair of Aces.

B. Ace/any face card combination 1.9%

C. Any combination of high cards 7.2%

(Ace or face cards) Slightly higher probability than getting any pair.

D. Combination of high cards suited 1.1%

(Ace and a face card suited)

Now we will look at the probability of opening with suited or connected cards:

A. Two suited cards 23%

B. Two suited and connected cards 3.9%

C. Two high suited and connected cards 1.2% (Jack and higher).

D. Two high connected cards 6.0% (cards higher than Ten)

Player Cards: Calculation Notes

Pairs:

A. Probability of a pair of Aces (4 / 52) × (3 / 51)

= .077 × .058 = .0045 = 0.45%

B. Any high pair (Aces, Kings, Queen, Jacks)

(16 / 52) × (3 / 51)

= .307 × .059 = .018 = 1.8%

C. Any pair higher than nines

(24 / 52) × (3 / 51)

= .461 × .059 = .027 = 2.7%

D. Any pair

1 × (3 / 51)

= .059 = 5.9%

High card combinations:

A. Ace/King combination

(8 / 52) × (4 / 51)

= .153 × .078 = .012 = 1.2%

B. Ace with any face card

(4 / 52) × (12 / 51)

= .076 × .255 = .019 = 1.9%

C. Any combination of high cards (Ace or face cards)

(16 / 52) × (12 / 51)

= .307 × .255 = .072 = 7.2%

D. Ace/face card combination suited

(4 / 52) × (3 / 51)

= .019 × .059 = .0011 = 1.1%

Suited or Connected Cards

A. Two suited cards 1 × 12/51 = 23.5%

B. Two suited and connected cards (Define connected as card next in sequence) 1 × 2/51 = 3.9%

C. Two high suited and connected cards 16/52 × 2/51 = 1.2%

Chapter Three

One Pair

Now the player cards have been dealt, and you have a high-card combination, what is the probability of getting one pair or two pair? What is the probability of hitting a set, a straight draw, or a flush draw on the flop?

From this point forward, we will always be discussing probabilities after the players have reviewed their hands. We will also analyze the probabilities at three different stages:

1. Before the flop (three community cards)

2. Before the turn (fourth community card)

3. Before the river (fifth community card)

Since these stages are the betting points and the cards are revealed at each stage, we will also discuss the payback ratio (pot-size-to-final-bet-size) to determine the mathematical feasibility of calling on the river.

Probability of One Pair

With each player receiving two cards, the probability of one player getting one pair is as follows:

1. At the flop 36.7%

2. At the turn 12.7%

3. At the river 13.0%

Total probability of getting one pair: 62.3%

As can be seen from above, the probability of matching one pair is very high, so if you are a player that awaits high-card combination to play, your probability of making a high pair is six hands out of ten.

For example, if you are playing an Ace/Queen opening hand, the probability of getting an Ace or Queen is as follows:

On the flop (6 outs / 50 cards) × 3 cards to be opened = 36.7%

On the turn (6 outs / 47 cards) × 1 card to be opened = 12.7%

On the river (6 outs / 46 cards) × 1 card to be opened = 13.0%

So the total probability of getting one pair is 36.7% + 12.7% + 13.0% = 62.3%.

Now you must note that the probability of matching one card only will be half of the above. So if you are playing a high card and a low card, such as Ace/Five or King/Four, the probability of matching the high card only is exactly half of the above.

Payback Information

Consider the scenario in which you are playing two high cards and you have not matched a pair through the turn, but you think you can win the hand with one pair (by matching on the river). The payback is as follows:

Playing an Ace/Queen hand, the probability of hitting an Ace or a Queen on the river is 13%.

Payback Ratio: (100 / 13) = 7.7:1

So if the bet is $10.00, the pot size should be a minimum of $77.00 to make calling mathematically feasible.

Summary

The probability getting one pair on the flop is 36.6%. If you have high cards in your hand and you did not match a pair on the flop, the probability now of getting one pair on the turn or river is around 25%.

Remember, the total probability of getting one pair is 62.3%, meaning that just over six times out of ten, your opponent will also make a pair.

If there are a total of five players in a hand (you plus four opponents), and assuming that there are eight different cards dealt, the probability of someone having a pair is 8 out of 13. So with five players, (8 / 13) = 61.5% that one of your opponents has made a pair.

Your probability of catching a pair on the river is 13%, and the payback is approximately 7.7:1. This means that the pot has to be more than eight times the size of the bet to make calling feasible.

One Pair: Calculation Notes

Probability of one pair on the flop: three opportunities, six outs for each opportunity:

(6 / 50) + (6 / 49) + (6 / 48)

= .120 + .122 + .125 = .367 = 36.7%

On the turn, one opportunity:

6 / 47

= .127 = 12.7%

On the river, one opportunity:

6 / 46

= .130 = 13.0%

Chapter Four

Two Pair

There are many ways of getting two pair.

A. On the flop

The probability of matching two pair right at the flop is 2.23%.

If you are playing a hand of Ace/Queen, just for example, the probability of matching two pair at flop is as follows:

You can match the first pair (either Ace or Queen, six outs) on the first card, then you have three outs, but two opportunities to match the second pair.

(6 / 50) × ((3 / 49) + (3 / 48)) = 1.47%

If you did not match either card on the first card, then you have to match the first pair on the second card and second pair on the third card. You have three outs to be matched on the second card and then also three outs on the third card.

1 × (3 / 49) × (3 / 48) = 0.75%

Add those together. Total probability = 2.23%

B. One pair on the flop, the second on the turn or river

The probability of matching one pair at the flop, as discussed in Chapter Three, is 36.7%.

Second pair on the turn = 6.4% Second pair on the river = 6.5%

So if you had one pair on the flop, the total probability of getting a second pair is at 12.9%. Your payback number will be 7.5:1.

C. No pair after the flop

Probability of getting first pair on the turn = 12.5% Probability of getting second pair on the river = 6.5%

Total probability of getting two pair after flop = 0.8%

D. The board pairing up on the flop

In this scenario, the first card could be any card, and there are two opportunities to match it. If the second card does not match the first card, there is still one opportunity to match on the third card. The calculations are as follows:

1 × ((3 / 49) + (3 / 48)) = 12.3%

1 × 1 × (3 / 48) = 6.25%

Total probability = 18.5%

So the probability of any pair showing up on the flop is 18.5%.

E. The board pairing up on the turn or the river

What is the probability of the board pairing up on the turn or river? Sometimes this is very important, such as if you are holding a set and need the board to pair up to make a full house.

Any three cards come on the flop. The probability of matching a pair:

On the turn, you have nine outs to match any of the three cards on the flop.

9 / 47 = 19.1%

On the river, now there are twelve outs to match any card on the board.

12 / 46 = 26.0%

Total probability = 45.1%

So just remember, there is a 45% chance that the board will pair up after the flop.

Probability of Two Pair

A. The probability of getting two pair right on the flop is very low, 2.25%. However, there is a high probability that this hand will improve dramatically (see the chapter on full houses).

B. The probability of getting one pair on the flop is 36.5%. (If you are lucky and have indeed flopped a pair). The probability of getting a second pair on the turn or river is at 12.5%.

C. If you have a high pair in your hand, the probability of a pair showing up on the flop is 18.5%.

Above all else, remember that the probability of the board pairing up after the flop is very high, about 45% of the time.

Two Pair: Calculation Notes

A. Two pair on the flop

You match either one of your player cards on the first card, then you have two opportunities to match the second card.

(6 / 50) × ((3 / 49) + (3 / 48)) = 1.47%

If you miss the first card on the flop, then you must match the first pair on the second card and second pair on the third card.

1 × (6 / 49) × (3 / 48) = 0.076%

Total probability = 2.23%

B. One pair on the flop, second pair on the turn or river

If you have one pair on the flop, the probability of a second pair comes with two chances.

Second pair on the turn: 3 outs / 47 cards = 6.4% Second pair on the river: 3 outs / 46 cards = 6.5%

Chapter Five

Three of a Kind

There are two ways to get three of a kind, also called a set:

A. Getting a set starting with a pair in your opening hand

On the flop = 12.2% On the turn = 4.2% On the river = 4.3%

The total probability is 20.7%

In summary, just remember that if you start with a pair in your hand, the probability of hitting a set on the flop is 12.2%, and the probability of hitting a set overall is 20.7%.

In reality, in most situations if you're holding a low pair in your hand, it is hard to call on the turn or river because the payback number is so high. The probability of making a set on the turn is 8.5% because you have the chance from the turn card and another chance from the river card.

(Continues...)

Excerpted from TO THE RIVERby Sam Habash Copyright © 2011 by Sam Habash. Excerpted by permission of iUniverse, Inc.. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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