The Easiest Way to Understand Math in Poker

The Easiest Way to Understand Math in Poker

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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.

New Poker Player Guide to Outs and Pot Odds

To respond to a request on Twitter by JamesM_39:

What are outs in poker?

The number of cards you need to complete your draw or make a specific hand. For example, if you flop a flush draw you have 9 outs because there are nine cards in your suit that are potentially left in the deck that can make your flush. There may not be 9 outs since one or more players may have received your desired suited cards as their hole cards.

Why is it important to know the number of outs?

It is important to know the outs you have since you want to know the probability of making your hand on the the turn and or the river. And you want to compare that outcome with the pot odds are you are getting.

Here is a quick way to handle your outs and the probabilities:
1. If you are only going to see one more card, take the number of outs and multiply it by 2 to get the percentage. With that flush draw, you have 9 outs times 2 or an 18% probability of making your hand on the turn.

2. If you are going to see both the turn and river cards, take the number of outs and multiply it by 4 to get the percentage. With that flush draw, you have 9 outs times 4 or an 36% probability of making your hand on the turn and/or the river.

Now that you can quickly calculate outs into probabilities, let's look at pot odds.

What are pot odds?

Pot odds are simply the amount of money you are betting compared to the amount of money you can win that is already in the pot.

For example, if you are betting $50 to win a pot that has $100, you are betting $1 to win $2. When you convert that into percentages you divide 1 by 3, which is 33%. The reason you divide by 3 and not 2 is that it is like there are three possible outcomes, where one time you win, and the other two times you lose.

Therefore, if you are on a flush draw on the flop and plan to see the river card, you know you have 9 outs or a 36% probability of making your hand. If you had to bet $50 to win $100 in this situation, you would only be getting pot odds of 33%--which is lower than the odds you need of actually making your hand. Therefore, all things being equal, you should fold if this is the only factor to take into your decision.

Odds into Probabilities

Here is a simple chart of the number of Outs and the Estimated Probabilities if you stay to the river. Again, it is simply the number of outs multiplied by 4.
1=4%
2=8%
3=12%
4=16%
5=20%
6=24%
7=28%
8=32%
9=36%
10=40%
11=44%
12=48%
13=52%

Therefore, now all you have to do is to know how much you are going to bet and compare it to the amount you can win in the pot, in order to get another percentage for your pot odds.

If your pot odds percentage is greater than the number of outs percentage you should stay in the hand and at least call your opponent's bet. Of course, there may be other information you have, which indicates a fold is in order. For example, another player groans when two cards to a flush flops and you hear him whisper to his neighbor, "I mucked a flush draw."

Don't laugh. That situation recently happened to me, and it let me get away from my hand.

I hope this was easy to understand. I know some players don't like to do the math, but poker math is not that difficult. Good luck!

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.

How to take advantage of poker odds to win big pots

You probably know what pot odds and implied odds are in poker, but do you know how to take full advantage of this knowledge in a no limit poker tournament?

As a reminder, pot odds are the ratio of the size of the pot to the size of the bet. If the pot has $400 in it, and your opponent bets $400, the pot is $800. Since you need to call with $400, the pot odds are $800 to $400 or 2-1.

Implied odds are based on the amount you believe you may win at the end of the hand, if you make your hand and win.

For example, the pot has $1,000 after your opponent bets $500 on the flop. You have a straight draw, and the pot odds are $1,000 to $500 or 2-1. Which is not favorable for drawing to a straight.

However, if you hit your straight on the next card, you believe that you may end up winning $4,000. Therefore, your estimate of your implied odds is $4,000 to $500 or 8-1 and you can call the bet.

While it's important to know these odds, it's even more important to know how to take advantage of the odds. Here's an example:

You are in a no limit poker tournament. It is the first hand with the blinds $25-$50. Everyone starts with $4,000 You are in the big blind.

The player under the gun raises to $150. Everyone folds to you. You have the pot odds of $100 to call the bet. Your pot odds are $225 to $100, or slightly over 2-1.

You look down at your cards and you have 8-7 suited. What should you do?

The correct play in this situation is to call the bet since you are getting excellent implied odds. If your opponent has a big hand like pocket Kings, and you hit your hand, you could win a much bigger pot.

What happens in this situation is that right before you call the raise, your opponent picks up his cards to look at them. By accident he flashes his cards to you. He has pocket Aces! Now what should you do? Does this change your original decision?

Not at all. Call the bet.

As a general rule, when you are in the big blind consider calling a bet if you are getting 2-1 pot odds or better, have much bigger implied odds (due to deep stacks) and have a drawing hand or better.