Chances Of Winning Poker Tournament

4/11/20224/11/2022

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Introduction

Rules

As daunting as it sounds, it is simply a tool that we use during the decision making process to calculate the Pot Odds in Poker and the chances of us winning the pot. Remember, Poker is not based on pure luck, it is a game of probabilities, there are a certain number of cards in the deck and a certain probability that outcomes will occur. Texas Hold'em Introduction Rules. A single 52-card deck is used. All cards count as its poker value. Aces may be high or low. One player is designated as the dealer, usually with a laminated marker. The odds won’t always fall in your favour, but over the long term, those Aces will win more than they lose against the 9s. Learning to win at poker is a long-term project that requires playing thousands and thousands of hands in a real game setting.

A single 52-card deck is used. All cards count as its poker value. Aces may be high or low.
One player is designated as the dealer, usually with a laminated marker. This person does not have to physically deal the game. However it is important that a symbolic dealer position rotate around the table.
The player to the dealer's left must make a 'small blind' bet. The player to the left of the small blind must make a 'big blind' bet. The amounts of both blinds should be specified in advance. The purpose of the blinds is to get the ball rolling with some money in the pot.
Two cards shall be dealt down to each player, starting with the person to the dealer's left.
The player to the left of the big blind must either call or raise the big blind bet. The play in turn will go around the table according to normal poker rules, which I assume the reader already knows. Table rules will specify any limits on the size or number of allowed raises.
The small blind may also raise the big blind. If nobody raises the big blind the player making the big blind has the option to raise his own bet. The term for this is the 'big blind option.'
Three community cards will be dealt face up in the center of the table. This is called the 'flop.'
Another round of betting will ensue, starting with the player to the dealer's left.
A fourth community card will be dealt face up in the center of the table. This card is called the 'turn.'
Another round of betting will ensue, starting with the player to the dealer's left. Generally the minimum bet is double the first two rounds of betting.
A fifth and final community card will be dealt face up in the center of the table. This card is called the 'river.'
Another round of betting will ensue, starting with the player to the dealer's left. The minimum bet is generally the same as the previous round.
Each player still in the game at the end will determine the highest poker value among his own two cards and the five community cards. It is NOT a requirement that the player use both of his own cards. The player with the hand of highest poker value shall win. Following are the hand rankings.
1. Straight flush: Five consecutive and suited cards. For example 5, 6, 7, 8, 9.
2. Four of a kind: Four cards of the same rank, plus any fifth card. For example Q, Q, Q, Q ,4.
3. Full house: Three of a kind and a pair. For example 6, 6, 6, J , J.
4. Flush: Any five cards of the same suit, except for a higher ranking straight flush. For example A, Q, 8, 4 , 3.
5. Straight: Five consecutive cards, except for a higher ranking straight flush. For example 8, 9, 10, J, Q.
6. Three of a kind: Three cards of the same rank, plus any other two cards. For example 5, 5, 5, Q ,2 .
7. Two pair: Two pairs, plus any fifth card. For example 8, 8, 2, 2 ,Q .
8. Pair: A pair and any other three cards. For example 7, 7, 2, 5 ,A .
9. ? High: Any five cards that do not form any higher poker hand. A king high hand for example might be K, Q, 7, 5 ,4 .
If two or more players have poker values of the same rank then the individual cards will be used to break the tie. If necessary all five cards will be considered.
I get asked a lot whether the two unused cards in a player's hand are used to break a tie. The answer is a firm NO. The two unused cards do not matter.
If a new player arrives at the table he should either wait for the big blind position or put up an amount equal to the big blind, amounting to a call of the big blind.
If a bet is made after another player runs out of money, then a separate pot is created. The player that ran out of money is not eligible to win the second pot. If more than one player runs out of money then multiple separate pots can be created.
In formal games players may not bet with cash or buy chips with cash in the middle of a hand.
There are numerous rules of etiquette, which I won't get into.
There house may set the betting rules. There are three main types. A 'structured' game features raises of specified amounts. For example a '3/6 game' would mean that raises after the deal and flop are $3, and after the turn and river are $6. There is usually a limit to the number of raises a player may make, typically three. A 'pot limit' game has structured minimum raises but the maximum raise may be anything up to the amount in the pot at the time the raise is made. A 'no limit' game also has structured minimum raises but there is no maximum raise.

Examples

Example 1

Board: A, 2, 4, 5, 6
Player 1: J, 6
Player 2: 7, Q

Player 1 wins. Both have an ace high flush, so the second highest card is considered. Player 1's jack beats player 2's 7. The only way to have a flush tie is if the flush is entirely on the board and no hole cards are higher than the lowest card on the board in the same suit.

Example 2

Board: J, A, 7, 5, 6
Player 1: 2, J
Player 2: 10, J

Player 2 wins. Both have a pair of jacks so the singletons are considered. High highet singleton in both hands is an ace so the second highest singleton is considered. Player 1's second highest singleton is a 7, compared to player 2's 10. A 10 beats a 7 so player 2 wins.

Example 3

Board: A, A, K, Q, J
Player 1: Q, J
Player 2: Q, 2

Tie. Both have a two pair of aces and queens, with a king singleton. Some people incorrectly believe that in such cases the unused cards are considered, in this case player 1's pair of jacks beating player 2's jack/2. Only the top five cards matter. The jacks and deuce are irrelevant.

One of the most important aspects of Texas Hold'em is the value of each two-card hand before the flop. The decision of how to play your first two cards is something you face every hand, and the value of your first two cards is highly correlated to your probability of winning.

The following table shows my power rating for each initial 2-card hand in a 10-player game. The numbers are on a 0 to 40 scale. Basically, you should only play hands that are dark green, blue, or purple. Of course you should be more be more liberal in late position and picky in early position. If forced I would say you should need 10 points in late position and 19 points in early position to call the big blind. If your table is loose, as if often the case online, you can play a bit looser yourself.

Use the top table if you have a pair, the middle table if your cards are suited, and the bottom table if your cards are unsuited. Except for a pair,look up your high card along the left and your low card along the top.

Following are the links to my tables of the value of each intial hand according to the number of players. The 10-player section explains the methodology for creating the table table.

Pot Odds

The following table shows the probability of making various hands after the flop and the correct 'pot odds.' The pot odds are the breakeven ratio of money in the pot to the amount you have to bet for the player to be indifferent about calling, assuming the player would definitely win if he makes the hand (a big if) and there are no additional bets (another big if). This table is a good starting point the player should make mental adjustments for the probability of winning without making the hand, losing with making the hand, and expected future bets. The odds of a two pair improving to a full house are the same as those for four to an inside straight.

Pot Odds — After Flop

Hand	Probability of Making Hand	Pot Odds
Four to a flush	34.97%	1.86
Four to an outside straight	31.45%	2.18
Four to an inside straight	16.47%	5.07

The next table shows the pot odds after the turn.

Pot Odds — After Turn

Hand	Probability of Making Hand	Pot Odds
4 to a flush	19.57%	4.11
4 to an outside straight	17.39%	4.75
4 to an inside straight	8.70%	10.50

Hand Strength Calculator

I'm proud to present my new and improved Poker Odds Calculator. Enter any situation in Texas Hold 'Em, and it will tell you the probability of each possible outcome.

Poker Tournament Calculator

My Poker Tournament Calculator will determine each player's probability, for up to nine players, of finishing in each place, and his expected share of any prize pool, assuming equal skill among all players. It produces the same results as what is known as the Independent Chip Model.

Internal Links

Pinapple — Strategy and analysis of which card to discard before the flop.
Bad Beat Jackpots: What is the Probability of Hitting one?
Texas Hold 'Em Dominated Hand Probabilities: What is the probability one of your opponents has similar, and better, hole cards than yours?

Written by:Michael Shackleford

This is a very important lesson and can also be quite intimidating to a lot of people as we are going to discuss Poker Math!

But there is no need for you to be intimidated, Poker Maths is very simple and we will show you a very simple method in this lesson.

You won’t need to carry a calculator around with you or perform any complex mathematical calculations.

What is Poker Math?

As daunting as it sounds, it is simply a tool that we use during the decision making process to calculate the Pot Odds in Poker and the chances of us winning the pot.

Remember, Poker is not based on pure luck, it is a game of probabilities, there are a certain number of cards in the deck and a certain probability that outcomes will occur. So we can use this in our decision making process.

Every time we make a decision in Poker it is a mathematical gamble, what we have to make sure is that we only take the gamble when the odds are on in our favour. As long as we do this, in the long term we will always come out on top.

When to Use Poker Maths

Poker Maths is mainly used when we need to hit a card in order to make our hand into a winning hand, and we have to decide whether it is worth carrying on and chasing that card.

To make this decision we consider two elements:

How many “Outs” we have (Cards that will make us a winning hand) and how likely it is that an Out will be dealt.
What are our “Pot Odds” – How much money will we win in return for us taking the gamble that our Out will be dealt

We then compare the likelihood of us hitting one of our Outs against the Pot Odds we are getting for our bet and see if mathematically it is a good bet.

The best way to understand and explain this is by using a hand walk through, looking at each element individually first, then we’ll bring it all together in order to make a decision on whether we should call the bet.

Consider the following situation where you hold A 8 in the big blind. Before the flop everyone folds round to the small blind who calls the extra 5c, to make the Total pot before the Flop 20c (2 players x 10c). The flop comes down K 9 4 and your opponent bets 10c. Let’s use Poker Math to make the decision on whether to call or not.

Poker Outs

When we are counting the number of “Outs” we have, we are looking at how many cards still remain in the deck that could come on the turn or river which we think will make our hand into the winning hand.

In our example hand you have a flush draw needing only one more Club to make the Nut Flush (highest possible). You also hold an overcard, meaning that if you pair your Ace then you would beat anyone who has already hit a single pair on the flop.

From the looks of that flop we can confidently assume that if you complete your Flush or Pair your Ace then you will hold the leading hand. So how many cards are left in the deck that can turn our hand into the leading hand?

Flush – There are a total of 13 clubs in the deck, of which we can see 4 clubs already (2 in our hand and 2 on the flop) that means there are a further 9 club cards that we cannot see, so we have 9 Outs here.
Ace Pair – There are 4 Ace’s in the deck of which we are holding one in our hand, so that leaves a further 3 Aces that we haven’t seen yet, so this creates a further 3 Outs.

So we have 9 outs that will give us a flush and a further 3 outs that will give us Top Pair, so we have a total of 12 outs that we think will give us the winning hand.

So what is the likelihood of one of those 12 outs coming on the Turn or River?

Professor’s Rule of 4 and 2

An easy and quick way to calculate this is by using the Professor’s rule of 4 and 2. This way we can forget about complex calculations and quickly calculate the probability of hitting one of our outs.

The Professor’s Rule of 4 and 2

After the Flop (2 cards still to come… Turn + River)
Probability we will hit our Outs = Number of Outs x 4
After the Turn (1 card to come.. River)
Probability we will hit our Outs – Number of Outs x 2

So after the flop we have 12 outs which using the Rule of 4 and 2 we can calculate very quickly that the probability of hitting one of our outs is 12 x 4 = 48%. The exact % actually works out to 46.7%, but the rule of 4 and 2 gives us a close enough answer for the purposes we need it for.

If we don’t hit one of our Outs on the Turn then with only the River left to come the probability that we will hit one of our 12 Outs drops to 12 x 2 = 24% (again the exact % works out at 27.3%)

To compare this to the exact percentages lets take a look at our poker outs chart:

After the Flop (2 Cards to Come)			After the Turn (1 Card to Come)
Outs	Rule of 4	Exact %	Outs	Rule of 2	Exact %
1	4 %	4.5 %	1	2 %	2.3 %
2	8 %	8.8 %	2	4 %	4.5 %
3	12 %	13.0 %	3	6 %	6.8 %
4	16 %	17.2 %	4	8 %	9.1 %
5	20 %	21.2 %	5	10 %	11.4 %
6	24 %	25.2 %	6	12 %	13.6 %
7	28 %	29.0 %	7	14 %	15.9 %
8	32 %	32.7 %	8	16 %	18.2 %
9	36 %	36.4 %	9	18 %	20.5 %
10	40 %	39.9 %	10	20 %	22.7 %
11	44 %	43.3 %	11	22 %	25.0 %
12	48 %	46.7 %	12	24 %	27.3 %
13	52 %	49.9 %	13	26 %	29.5 %
14	56 %	53.0 %	14	28 %	31.8 %
15	60 %	56.1 %	15	30 %	34.1 %
16	64 %	59.0 %	16	32 %	36.4 %
17	68 %	61.8 %	17	34 %	38.6 %

As you can see the Rule of 4 and 2 does not give us the exact %, but it is pretty close and a nice quick and easy way to do the math in your head.

Now lets summarise what we have calculated so far:

We estimate that to win the hand you have 12 Outs
We have calculated that after the flop with 2 cards still to come there is approximately a 48% chance you will hit one of your outs.

Now we know the Odds of us winning, we need to look at the return we will get for our gamble, or in other words the Pot Odds.

Pot Odds

When we calculate the Pot Odds we are simply looking to see how much money we will win in return for our bet. Again it’s a very simple calculation…

Pot Odds Formula

Pot Odds = Total Pot divided by the Bet I would have to call

What are the pot odds after the flop with our opponent having bet 10c?

Total Pot = 20c + 10c bet = 30 cents
Total Bet I would have to make = 10 cents
Therefore the pot odds are 30 cents divided by 10 cents or 3 to 1.

What does this mean? It means that in order to break even we would need to win once for every 3 times we lose. The amount we would win would be the Total Pot + the bet we make = 30 cents + 10 cents = 40 cents.

Bet number	Outcome	Stake	Winnings
1	LOSE	10 cents	Nil
2	LOSE	10 cents	Nil
3	LOSE	10 cents	Nil
4	WIN	10 cents	40 cents
TOTAL	BREAKEVEN	40 cents	40 cents

Break Even Percentage

Now that we have worked out the Pot Odds we need to convert this into a Break Even Percentage so that we can use it to make our decision. Again it’s another simple calculation that you can do in your head.

Break Even Percentage

Break Even Percentage = 100% divided by (Pot odds added together)

Let me explain a bit further. Pot Odds added together means replace the “to” with a plus sign eg: 3 to 1 becomes 3+1 = 4. So in the example above our pot odds are 3 to 1 so our Break Even Percentage = 100% divided by 4 = 25%

Note – This only works if you express your pot odds against a factor of 1 eg: “3 to 1” or “5 to 1” etc. It will not work if you express the pot odds as any other factor eg: 3 to 2 etc.

So… Should You call?

So lets bring the two elements together in our example hand and see how we can use the new poker math techniques you have learned to arrive at a decision of whether to continue in the hand or whether to fold.

To do this we compare the percentage probability that we are going to hit one of our Outs and win the hand, with the Break Even Percentage.

Should I Call?

Call if…… Probability of Hitting an Out is greater than Pot Odds Break Even Percentage
Fold if…… Probability of Hitting an Out is less than Pot Odds Break Even Percentage

Our calculations above were as follows:

Probability of Hitting an Out = 48%
Break Even Percentage = 25%

If our Probability of hitting an out is higher than the Break Even percentage then this represents a good bet – the odds are in our favour. Why? Because what we are saying above is that we are going to get the winning hand 48% of the time, yet in order to break even we only need to hit the winning hand 25% of the time, so over the long run making this bet will be profitable because we will win the hand more times that we need to in order to just break even.

Hand Walk Through #2

Lets look at another hand example to see poker mathematics in action again.

Before the Flop:

Blinds: 5 cents / 10 cents
Your Position: Big Blind
Your Hand: K 10
Before Flop Action: Everyone folds to the dealer who calls and the small blind calls, you check.

Two people have called and per the Starting hand chart you should just check here, so the Total Pot before the flop = 30 cents.

Flop comes down Q J 6 and the Dealer bets 10c, the small blind folds.

Do we call? Lets go through the thought process:

How has the Flop helped my hand?
It hasn’t but we do have some draws as we have an open ended straight draw (any Ace or 9 will give us a straight) We also have an overcard with the King.

How has the Flop helped my opponent?
The Dealer did not raise before the flop so it is unlikely he is holding a really strong hand. He may have limped in with high cards or suited connectors. At this stage our best guess is to assume that he has hit top pair and holds a pair of Queens. It’s possible that he hit 2 pair with Q J or he holds a small pair like 6’s and now has a set, but we come to the conclusion that this is unlikely.

How many Outs do we have?
So we conclude that we are facing top pair, in which case we need to hit our straight or a King to make top pair to hold the winning hand.

Open Ended Straight Draw = 8 Outs (4 Aces and 4 Nines)
King Top Pair = 3 Outs (4 Kings less the King in our hand)
Total Outs = 11 Probability of Winning = 11 x 4 = 44%

What are the Pot Odds?
Total Pot is now 40 cents and we are asked to call 10 cents so our Pot odds are 4 to 1 and our break even % = 100% divided by 5 = 20%.

Decision
So now we have quickly run the numbers it is clear that this is a good bet for us (44% vs 20%), and we make the call – Total Pot now equals 50 cents.

Turn Card

Turn Card = 3 and our opponent makes a bet of 25 cents.

After the Turn Card
This card has not helped us and it is unlikely that it has helped our opponent, so at this point we still estimate that our opponent is still in the lead with top pair.

Outs
We still need to hit one of our 11 Outs and now with only the River card to come our Probability of Winning has reduced and is now = 11 x 2 = 22%

Pot Odds
The Total Pot is now 75 cents and our Pot odds are 75 divided by 25 = 3 to 1. This makes our Break Even percentage = 100% divided by 4 = 25%

Decision
So now we have the situation where our probability of winning is less than the break even percentage and so at this point we would fold, even though it is a close call.

Summary

Well that was a very heavy lesson, but I hope you can see how Poker Maths doesn’t have to be intimidating, and really they are just some simple calculations that you can do in your head. The numbers never lie, and you can use them to make decisions very easy in Poker.