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.

Unexpected Value: A New Idea in Tournament Poker

Day 55, the Countdown Continues

"The mathematically correct play is not always the best play."
-Chip Reese

What is unexpected value? Unexpected value is making a fold even though you have positive expected value, since you know you will get more value out of a later poker situation.

Example:

It was down to 18 players in a MTT. There were 9 players at my table and I was on the button. A tight player raised 3x's the big blind. It was folded to me. I found pocket Jacks. A call made no sense since it committed half my stack. It was a situation where if I moved all-in, the raiser would call me since he had committed about 40% of his chip stack.

Since I knew my opponent was tight, he was pot committed, and 65% of the time the flop alone will have one card higher than my Jack, I made the unusual play. I folded.

I believed the unexpected value of folding was greater than the expected value I may have enjoyed at the moment.

It ended up being the right thing to do since I was able to take my low chip stack and finish 4th.

Let's take this concept one step further:

You need chips to win a tournament, and our goal is to win.

Expected value is a solid cash game concept since it is about how mathematically in the long term you will be ahead.

Expected value does not play out in all tournament decisions because a tournament is one event that is limited in time and where the value of a chip changes over time.

Another example of unexpected value is a player who views the size of the pot and decides to throw in the last of his chips since the pot is so big. Well, one of the top tournament players mentioned that he used to make this mistake and get knocked out. Now, he will save those chips and use them to build his stack back up.

While he didn't give this idea a name, I think it is my unexpected value concept.

Another example is the way Phil Hellmuth plays no limit tournaments. Phil has cashed more often than any other player in these events. One thing I read is that Phil does not always make a call when he is on a draw. I wonder if it is because of the odds in the situation, or because Phil knows he has the edge by waiting for the right situation to accumulate chips.

One final example is where I took the above quote from the late Chip Reese. He told a story about how was much better than his opponent, and he risked a large percentage of his chips against this man since he knew he had a slight edge. He was right but he ended up losing the hand. That's when he said "The mathematically correct play is not always the best play." (Anyone recall the poker book I read that in?)

Perhaps that sums up the concept of unexpected value. Unexpected value is when the mathematically correct play is not the best play.

What do you think?

Expected Value in Poker: A Lousy Concept for the WSOP?

Day 56 The Countdown Continues...

What is expected value?

From wikibooks.org:

The expected value (EV) or expectation of a wager is how much you will win in the "long run" by making that wager.

For example, this situation comes up often in Texas hold'em: a player has four hearts on the turn, and needs to hit a heart on the next and final card to make a flush. He is absolutely certain he will win if he hits the flush and in so doing the board does not pair, because the board allows no possibility for a full house or better hand, and his flush will be the nut flush. Even if the board pairs, he has a good chance of winning, so he need not worry about other hands that much. The probability of catching the heart is about one in five. Therefore, he can expect to lose most of the time. However, the idea of continuing cannot be dismissed out of hand unless we know what his expectation is.

If it will cost $5 to call a bet and the pot, including other bets and calls, is $50, our player actually has a very positive expectation and should pay to see the next card. The chance of hitting his hand is only one in five, the pot is ten times the bet size. The player expects to win on average two dollars for every dollar invested every time this situation occurs.

Expected Value: A Lousy Concept for Tournaments

The more I play tournaments the more I learn that expected value is a lousy concept for tournaments. Here are some of my reasons why:

1. It is based on the "long run."
Cash games are in the long run. A tournament is not the long run. It is one moment in time.

2. It is based on the value of the chips being equal.
In cash games, a buck is a buck. In a tournament, the value of your chips change based on the blinds. As the blinds increase, it is said your chips are "worth" less.

3. Tournaments provide more opportunities to accumulate chips, so why not play when you have the upper hand.

I recall Chip Reese's story on how he lost a big pot against a poor player even though he had a slight edge. He said that sometimes the mathematically correct play is not the right play. And I think this is what happens when you adhere to expected value in a poker tournament.

My new tournament poker concept: Unexpected Value

What is unexpected value? Unexpected value is making a fold even though you have positive expected value, since you know you will get more value out of a better poker situation.

I will give you an example from a MTT I was playing in last night. It was down to 18 players. There were 9 players at my table and I was on the button. A tight player raised 3x's the big blind. It was folded to me. I found pocket Jacks. A call made no sense since it committed half my stack. It was a situation where if I moved all-in, the raiser would call me since he had committed about 40% of his chip stack.

Since I knew my opponent was tight, he was pot committed, and 65% of the time the flop alone will have one card higher than my Jack, I made the unusual play. I folded.

It ended up being the right thing to do since I was able to take my low chip stack and finish 4th.

What do you think?

Please let me know your thoughts on unexpected value in poker:) Thanks!