The Easiest Way to Understand Math in Poker

The Easiest Way to Understand Math in Poker

Image by alist via Flickr

I have read quite a few poker books. For me, if I can learn just one concept in a poker book, I feel the investment is a good one.

I have been reading Swayne's Advanced Degree in Hold'em and he has an excellent section on poker math. I think it may be the best I have ever seen. Here is a summary of this section.

1. Odds

Let's start with odds. In poker, if the odds are 14 to 4 it is shown as 14:4. The left hand side of the 14:4, the 14, means how many times something won't happen and you compare it to the right side, the 4, which means how often something will happen. Said another way, for every 18 events (14+4), it won't happen 14 times and it will happen 4 times.

Key: The left hand side is how many times something won't happen: the right hand side is how often something will happen.

The odds of 14 to 4 and be simplified by dividing the right hand side number, the 4, into the left hand side number, the 14. So, 14 divided by 4 is 3.5. Therefore the odds of 14 to 4 are the same as 3.5 to 1 or 3 1/2 to 1.

2. A Poker Example of Odds

Let's say you have two clubs in your hand, and the flop has two clubs. What are the odds against you getting another club on the turn?

Simply, it is "how often you won't catch a club: how often you will will catch a club."

Since there are 3 cards on board and 2 in our hand, the number of cards we don't know are 47. The number of cards that will help are the 12 club cards subtracted by the four club clubs we know about it, or 9.

Therefore, there are 38 cards that won't help: 9 cards that will. Or, 38:9. To simplify, 38 divided by 9 is 4.2 to 1. The odds against you getting another club on the turn is 4.2 to 1.

3. Probability

Probabilities are expressed as a percent. To convert pot odds into probabilities is simple.

If the odds are 3:1, it means that for every 4 events, it will happen one time. 1 divided by 4 is .25.

.25 is the same as 25%. Therefore, you now know that if the odds are 3:1, it means that the probability is 25%.

Now that you understand odds and probabilities, let's move to pot odds and how to use pot odds.

4. Pot Odds

Pot odds are the $ odds which the pot is offering. The $ odds the pot is offering is the amount of money in the pot compared to the amount of money you must pay to stay in the hand.

Example:

There is $50 in the pot and you must bet $10 to say in. The pot is offering you $50 for your $10 bet, or said another way, you are risking $10 to win $50.

In odds for poker, this is $50:$10, or 50:10 or 5:1. Therefore, the pot is offering you 5:1.

5. Using Pot Odds

To help you decide whether to stay in or fold post flop, you want to know the odds the pot is offering you compared to the odds of making your hand.

Key: If your pot odds are greater than your odds of winning, you stay in the hand.

Another way to say it: if the pot is offering you more than your odds against (assuming that you will win when you make your hand), you stay in the hand.

Example:

Earlier we figured out the odds for catching the flush card on the turn as 4.2 to 1. If the pot is offering you $50 for your $10, then the pot is offering odds of 5 to 1. Since the pot odds are more than the odds of hitting your flush, you stay in the hand.

If there were only $30 in the pot, and you had to bet $10 to stay in the hand, you would only be getting 3:1. Since the pot is offering you less than your odds against, you fold.

This makes it simpler to understand when you are getting favorable odds to call a bet, and when you are not getting favorable odds. You want the pot to be giving you better odds than the odds against making your hand in order to make a favorable call.

Another concept in poker is expected value or EV. Let's see how that works.

6. Expected Value (EV)

Expected value means the probability of winning times the amount you can win minus the probability of losing times the amount you must bet to see if you win.

A positive expected value means that over the long run you will win more than you will lose.

Example:

Again, assuming we have flopped a flush draw, what is the probability of making a flush on the turn?

Probability of catching the card=9 cards divided by 47 cards or .19
Probability of not catching the card is 1.00-.19 or .81.

The EV=(.19)(pot size)-(.81)(amount you must bet)
The EV=(.19)($50)-(.81)($10) or $9.5 minus $8.1= +$1.40

You will notice that the size of pot and the amount you must bet is critical in this calculation.

A positive value is good. It means that over the long run, you will win an average of $1.40.

Of course, it does not mean that you will win every time, but as long as you base your play on positive EV, you will win long term.

Using EV is the same thing as using pot odds.

How to Make Tournament Poker Easier For You To Beat...Maybe? Yes, You Need Luck & Avoid Bad Beats

In the last post, I reviewed how to evaluate a poker situation based on the pot odds compared to the odds of winning in different heads-up poker situations.

Let's see how that works out and in order to make your future decisions easier.

1. Common All-in Hands
You raise with A-K or a small pair but a player moves all-in on you. What should you do?

A) You have A-K.
Common hands and odds of A-K versus:
Q-Q: 43%
6-6: 45%
9d-Td: 58%
A-A: 7%
If you have to put in 10k to win 20k or 1 to 2=33% pot odds, a call is usually a good idea. In fact, this is why it is so difficult to get away from A-K. The pot odds will look good enough to call.

B)You have a weak pair like 5's.
Common hands and odds:
A-K: 55%
9d-8d: 49%
K-T: 47%
9-9: 19%

If you have to put in 10k to win 30k or 1 to 3=25% pot odds, a call is usually a good idea. This is why players move all-in with small pairs, and why more and more players are not calling a raise with small pairs but re-raising pre-flop. Therefore, with a medium pair, it may be a good idea to usually re-raise pre-flop since you only have two outs on the flop to improve.

2. Common heads-up situations
There are common heads-up face-offs, where you should know the odds. When you are playing poker the approximate odds are often good enough.

A-K vs a pair is a pair is a coin flip. For example, against 2-2 it is 50%. Against a pair like J-J, it is 44%.

While pocket Jacks are often losers, pre-flop they are ahead of most hands. Like against A-Q, it is 57%.

A suited connector is a slight favorite against a low pair, like Ts-Js vs 2-2 is 53%. But, against, an overpair it will be a dog, like 9d-8d vrs JJ is 20%.

There are also those situations which players call dominated hands. For example, an A-T vrs A-9 is 67%. If you have the A-9, the pot odds are 1 to 2 or 33%.

3. Difficult In-game decisions.
The test of knowing your odds really comes down to those other times, where you have a good but not great hand, and you are trying to make the right decision. The right decision is not based on the outcome but based on the math.

K-T is a hand that is good but not great. Here are the odds against many others:
A-A: 14%
A-K: 26%
Q-Q: 29%
A-Q: 38%
A-9: 42%
A-5: 43%
5-5: 47%
Q-8: 64%
Q-T: 71%

If you have to put in 10K to win 16k or 1 to 1.6=38%, a call is the right play even if your opponent has A-Q! Of course, a decision to call is not that simple. You want to think about the probability your opponent is holding A-Q, as well as hands that are better like A-K and worse like a small pair such as 5-5.

4. Defending the Blind.
One of the things that Gus Hansen does a lot is calling raises in the BB and even the SB.

In general, you will be in a situation of getting 1 to 2 pot odds, or 33% in the BB. Suited connectors and connectors in general tend to be worth a call.

If you have a hand like 9-7 suited, your odds are better than you think:
Q-J: 38%
A-K: 40%
5-5: 49%
J-J: 20%

Even if they are not suited, your odds are about the same (as a rule, you lose about 3 percentage points):
Q-J: 34%
A-K: 37%
5-5: 46%
J-J: 16%

Maybe those calls by Gus are not that crazy after all.

5. End Game All-ins
When should you push all-in? You are bleeding chips with a low stack and blinds are increasing. Well, I have reviewed the concept of M in a prior post. But, you may be surprised at the range of hands you can be pushing given the odds.

Any Ace push. If you push with A-3 and get called, you are not in that bad a shape. If you are against, K-J, you are a slight favorite at 57%.

If you get called by premium pair, you will be a big dog like Q-Q at 28%--but that doesn't happen often AND you will still win about 1 in 4 times! You will probably be up against a pair like 5-5, and be at 30% or almost 1 to 2.

Any King push. Of course, the fact is that when you go card dead, you don't even see an Ace. What if you push with that weak King? If your opponent calls with A-Q, you are at 34%.

Any Queen push. Yesterday, I pushed with Q-8 on the cutoff and lost against A-3. I was at 42%. If my opponent had A-Q, I would have been dominated and therefore, at 25%.

6. Pot odds and the percentages.
Let me try to make your decision easier, by doing the pot odds math for you:

1 to 1: 50%
1 to 1.1: 48%
1 to 1.2: 45%
1 to 1.3: 43%
1 to 1.4: 42%
1 to 1.5: 40%
1 to 1.6: 38%
1 to 1.7: 37%
1 to 1.8: 36%
1 to 1.9: 34%
1 to 2: 33%
1 to 2.5: 29%
1 to 3: 25%
1 to 4: 20%

The more money you are risking compared to the size of the pot you can win, the better your hand vs hand odds need to be in order to make that call.

Conclusion and An Exercise

You should have a general idea about these situations and the respective odds. It is interesting in that there are more hands you should consider defending in your big blind, and there are more hands that you should be pushing with based on a low M.

One of the things that has gone away in no limit tournaments are the all-in specialists. These were players who just moved all-in pre-flop, with the objective of either building a big chip stack early or simply getting knocked out. I believe one day these type of players will come back into the game. Maybe you should be that player?!

The Exercise:
In Sklansky's book Tournament Poker for Advanced Players, he developed an all-in system. Here it is.

1. If someone else has raised in front of you, move all-in with pocket Aces, pocket Kings, or Ace-King suited. Otherwise fold.

2. If no one else has raised in front of you, move all in with any pair, any Ace-x suited, Ace-King, or two suited no gap connected cards, except 4-3 and 3-2.

You may want to try this in a few low buy-in events. It may be a great learning experience and you may even win an event!

NOTE: I believe he updated this system at a later date. If I find it, I will post. If you find it, please let me know what it is. Heck, you may want to refine it, and play that way. It may work! Thanks.

UPDATE: I did try Sklansky's System a few times. It was way too boring and it didn't work out at all. The move-in specialists that I played against many years ago were much smarter on when to make their all or nothing moves. The system doesn't take into account limpers, position, and especially your table image--as players slow play big hands to trap you. Don't waste your time with the Sklansky System.

This May Be My Most Important Post About Winning Tournament Poker Strategy

This may be my most important post because it should help you in 2 areas:
1. How to make a critical decision during a poker tournament based on the math.
2. How raising or re-raising pre-flop is better than calling or limping.
This winning approach is based on Gus Hansen's 2nd day at the Aussie Millions, which he won.

1. You Will Make Better Decisions If You Know The Math!

Hand 42
Blinds: 500/1000/100
Position: SB
Hand: 7h-7d
Chips: 68k

Button raises to 4000, and Gus re-raises to 12600. His opponent moves all-in for $45.3k.

Decision:
Step 1. What is Gus' winning chance?
He has to put in 33k to win 60k which is 1 to 1.8.
That means his opponent has a 1.8/2.8=64% winning chance,
A 36% winning chance for Gus.

Step 2: What is his chance of winning given common hand matchups?
Against an overpair he has an 18% chance
Against a hand like AQ or AK he has a 55% chance

Step 3: Which hand does your opponent have?
A raise and re-raise looks like an overpair, so Gus needs
more than a 36% chance of winning but it seems like he
only has an 18% chance.

Decision: Fold.

Hand 60
Blinds: 600/1200/200
Position: BB
Hand: Ah-7s
Chips: 143.4k

The player under the gun moves all-in for 11700. Everyone folds to Gus.

Decision:
Step 1. What is Gus' winning chance?
He has to put in 10.1k to win 15.7k which is 1 to 1.6.
That means his opponent has a 1.6/2.6=61% winning chance,
A 39% winning chance for Gus.

Step 2: What is his chance of winning given common hand matchups?
Gus isn't sure how strong his opponent is, since he
may be desperate.
If his opponent has a pair higher than 7's, he has only a
29% chance of winning.

Step 3: Which hand does your opponent have?
Since he isn't sure, he likes the 39% winning chance.

Decision: Call. He loses as his opponent wins with JJ.

Hand 63
Blinds: 800/1600/200
Position: 2nd
Hand: Ac-Qc
Chips: 133.5k

The player under the gun raises to 6100, Gus calls, the SB calls, but the BB moves all-in for 37600 more. It is folded to Gus. He knows the SB will fold.

Decision:
Step 1. What is Gus' winning chance?
He has to put in 37k to win 57k which is 1 to 1.6.
That means his opponent has a 1.6/2.6=61% winning chance,
A 39% winning chance for Gus.

Step 2: What is his chance of winning given common hand matchups?
As before, Gus isn't sure of his opponent's hand.

Step 3: Which hand does your opponent have?
But he thinks his hand is strong here.

Decision: Call. He wins as his opponent has AJ.

2. Raising versus Calling: This Time I'll Do The Math for Gus' Day 2 Play:

He raised or re-raised: 53 times
Number of times he won: 41 times
% Win Rate: 77%

He limped or called a bet: 19 times
Number of times he won: 10 times
% Win Rate: 53%

If a hand is good enough to call--especially those medium pairs--it may good enough to raise or even re-raise.

To get inside the mind of Gus Hansen, get his book Every Hand Revealed here. The above analysis is from reviewing his book.

Gus Hansen-winning playing style after 20 hands

Here is a review of hands 16-20, and the learning after 20 hands. I think we have an excellent portrait of Gus' winning playing style after just these hands. In addition, we start to get into the math that Gus uses to guide his decisions.

Hand 16:
Blinds 100/200/25
Position: Button
Hand Ad-6d
Chips: 26.1k

Gus is first and raises on the button to 600? Blinds fold.

Learning:
Steal on the button with a standard sized raise (although this hand is worth a raise).

Hand 17:
Blinds 100/200/25
Position: 2 off the button
Hand Jc-5c
Chips: 26.6k

Gus is first in and raises to 600. The button min raises to 1200, and Gus calls with great pot odds. Flop is Kc-9s-3c. Gus checks and opponent checks.

The turn is an 8c. Gus hit his flush. Gus bets out to 2000 and his opponent raises to 6000. His opponent either has KK or 99. Gus moves all-in for 19,375 and gets called. His opponent does have 99 for a set. River is a 4h. Gus doubles up.

Learning:
Bet out rather than check when you hit the flush on the turn.

Hand 18:
Blinds 150/300/25
Position: 2 off the button
Hand Kd-Qs
Chips: 53.2k

Player in early position limps, Gus limps (although he would normally raise), and blinds call as well. The flop is Kc-6h-2s.

Gus checks top pair after opponents check. Turn is 2h. BB bets to 500, limper raises to 2000 and only Gus calls. The river is a 3s.

Opponent bets 3000 into the 6000 pot, and Gus calls.

Learning:
It's okay to mix up your game and check a hand you would raise with pre-flop and bet with on the flop. But, realize, that your job of reading your opponents will be tougher since you sent out a message you were weak and not strong.

Hand 19:
Blinds 150/300/25
Position: 4th
Hand Ac-4c
Chips: 59.6k

Gus limps after under the gun limper. Blinds calls. Flop is 2h-5d-Kc. The limper bets 800 into 1425 pot.

Gus has a gut shot straight draw, backdoor flush draw, and Ace high. Since he has lots of chips he calls. Blinds fold. Turn is a Jc.

Opponent bets 2200, and has 7000 left. Gus has to call 2200 to win 5200 or a 30% winning chance. (Important: His winning chance is 2200 divided by the total of 2200+5200 or 7400.) Gus figures he has 10 outs of 44 remaining cards or 22.7%, meaning he should fold. But he calls given the implied odds.

The river is a 5h. Opponent bets and Gus folds.

Learning:
You can stretch your calls when you have a huge chip stack if you think your
implied odds are excellent.
Know your chance of winning given the bet size compared to the pot size, and then compare it to the number of cards you need to win compared to the umber of the remaining cards. If actual odds are higher than the cards percentage, you should fold.

Hand 20:
Blinds 150/300/25
Position: 2 off the button
Hand Jc-Tc
Chips: 55.7k

Gus opens for 1000 and only the BB calls. The flop is Jd-9d-8h. Gus bets 1200 and BB calls. The turn is a 4s.

There is 4800 in the pot and opponent has 12,000. If Gus bets 300 and his opponent calls, and the Kd hits on the river, Gus will have a tough decision. Gus does not like tough decisions. He puts his opponent on a draw and bets 13,000. His opponent folds.

Learning:
If you put your opponent on a draw, make an over-sized bet to get him to fold on the turn (putting him all-in).

Learning after 20 hands:

Limp in with suited connectors in early rounds to risk few chips to win a big pot.
Fold if you may be drawing dead when three cards of the same suit flop and you have a mid-level club.
First in on the SB raise.
Limp in back position with suited connectors, and call a raise behind you if the pot odds are big enough (over 3 to 1).
On paired flops, make a half pot sized bet against the blind limpers.
Raise with a mediocre hand first in, when in late position
Use a C-bet heads-up in position with more than half sized bet.
In BB call a SB raise first in that gives you 2-1 odds with suited connectors.
When hit a pair on flop, check raise to determine if you have best hand.
Raise from SB first in with suited connectors.
Call a raise HU if odds are right (over 3 to 1).
Just call A-J offsuit when there is an early position raise and one caller.
Raise under the gun with K-Q offsuit.
If c-bet gets called and turn is not a scare card, check your hand.
Call with suited connectors when the pot odds are right (2.5 to 1).
Cut-off is a good place to try to steal, but if get raised just fold.
On button, raise more than the size of the pot to try to take down a pot with all limpers.
Don't risk everything when you only have 2nd pair, if an opponent bets out and you are going to have to guess if you are beat or not. Wait for a better opportunity.
Raising with K-9 suited in early position is an acceptable move, especially when the table has shown a tendency to limp and fold to raises pre-flop.
Calling a raise with a flush draw on the flop is a good play with over 4 to 1 odds. (1800 to win 7550)
An aggressive table image can get opponents to make big mistakes against you.
Calling a re-raise with medium pairs is an acceptable play given the implied odds.
Steal on the button with a standard sized raise (although this hand is worth a raise).
Bet out rather than check when you hit the flush on the turn.
It's okay to mix up your game and check a hand you would raise with pre-flop and bet with on the flop. But, realize, that your job of reading your opponents will be tougher since you sent out a message you were weak and not strong.
You can stretch your calls when you have a huge chip stack if you think your
implied odds are excellent.
Know your chance of winning given the bet size compared to the pot size, and then compare it to the number of cards you need to win compared to the umber of the remaining cards. If actual odds are higher than the cards percentage, you should fold.
If you put your opponent on a draw, make an over-sized bet to get him to fold on the turn (putting him all-in).