# Poker Maths – Definitive Guide (2023)

• 32 mins read

We make the key poker maths concepts really simple. Learn about pot odds, blockers, combinatorics, minimum defence frequency, pot equity, expected value, and SPR.

• Learn poker maths with our simple easy to understand explanations.
• You should not play another hand of poker, without understanding these vital poker maths concepts.
• Start reading our definitive guide to all the poker maths you need to know, all in one place.
Jump straight to...

## 1. Pot Odds Poker Maths

Pot Odds are of the most important poker maths you will have to do, before making strategy decisions in a hand. The less you are risking, compared to your possible reward, the better.

### How to calculate pot odds

The ratio of the current pot size compared to the cost of a possible call, is the pot odds.

Just ask yourself 2 questions:

1. how much is already in the pot, before I call?
2. how much do I have to put in to make the call?

e.g. Lets say the pot size going into the river is \$200. You are heads up with your opponent. You each have \$200 left in your stacks, and your opponent decides to go all-in. What are your pot odds?

The answers to the 2 questions above are:

1. \$400. That’s the \$200 originally in the pot, PLUS the \$200 your opponent just bet.
2. \$200

So the answer is 400:200 or 2:1

### Convert to breakeven %

Pot odds of 2:1 means calling if you think you have the best hand more than 1/3 of the time. Notice, in this example you should be calling even if you expect to lose the hand the majority of times.

We use the following equation to work this out:

Amount to call
————————————————————————————————
(Amount to call ➕ Amount in pot before we call)

Try this mental shortcut when you see pot odds of x:y. Put y as the number at the top of the fraction, and x+y as the number of the bottom.

We recommend you think about this in the following way. If I choose to call, what % is the call of the new total pot?

## 2. Blockers Poker Maths

In poker, blockers & unblockers are pocket cards in your hand that change the probability of your opponent(s) holding part(s) of their possible range.

Blockers, are pocket cards in your hand that reduce the probability of your opponent(s) holding part(s) of their possible range. For example if you hold the A♠️3♦️ in your hand, and there are three spades on the board on the river, you block the nut flush.

Unblockers, are pocket cards in your hand that do not reduce the probability of your opponent(s) holding part(s) of their possible range. For example, imagine if you had a small pocket pair (e.g. 22) on the button. Your extremely value heavy opponent opened UTG on a full ring table. You might be unblocking their entire opening range.

It’s not just your pocket cards. Cards on the board, can also have blocking or unblocking effects on your opponent’s range. However, it is often more useful to have the blockers or unblockers in your pocket cards. This is because your opponent does not know you have this information.

Blockers, and unblockers, become more telling the less wide your opponent’s range. Imagine you open UTG on a full ring table. An extremely value heavy (and usually passive) opponent on the button raises. You might give them a range of QQ+ and AKs. But, if you had AA, this eliminates a massive part of their range.

### When are blockers & unblockers useful in game?

#### If facing a bet

• When continuing versus a bet, ideally we would want:
• to be blocking our opponent’s value range, and
• unblocking their bluff range.
• When attempting to catch a bluff (bluff-catching) it is good if:
• your pocket cards unblock your opponent’s bluff range, and
• block his value range.

#### If contemplating a bet yourself

• Ask yourself which parts of your opponent’s range you want to target? Are you blocking or unblocking these parts – does this increase or decrease your willingness to bet?
• When we are attempting to bluff, we want to block the hands our opponent will continue with. When we are attempting to bet for value, we want to unblock these hands.

#### Important Warning about blockers & unblockers

Blockers and unblockers should only be one part of your overall strategy.

#### Blockers and Poker Maths

You have 2 known cards in your hand – therefore your opponents cannot have these in there hand. These cards could be blockers to key hands in your opponent’s range. This information can be crucial.

##### Example

Let’s say you get A♥️K♠️ preflop, and you open UTG on a 9 handed table. A straightforward tight passive villain raises you next to act. You feel that he would only do this with AA, KK, QQ, and AKs. How many combos does he have of each?

Without any blockers, you would conclude he has:

• AA = 6 combos
• KK = 6 combos
• QQ = 6 combos
• AK suited = 4 combos
• (total = 22 combos)

So, for example let’s say instead you had opened JJ UTG and faced the same action. This time you have no blockers to range you are assigning villain. You conclude that 18 out of 22 times (when he holds AA, KK, or QQ), you are way behind. In these cases you have around 20% equity. 4 out of 22 times (when he holds AK suited) it’s a coin-flip (you have around 50% equity). Combining this information with pot odds, implied odds, SPR etc. you can make a decision about how to proceed.

With your A♥️ blockers and K♠️ blocker, his possible combinations change to:

• AA = 3 combos
• KK = 3 combos
• QQ = 6 combos
• AK suited = 2 combos
• (total = 14 combos)
##### Suited combos

It is extremely intuitive how many suited combos remain when you hold blockers. Each hand type can only have 4 suited combos. If you hold 1 blocker, 3 suited combos remain. If you hold 2 blockers, 2 suited combos remain. For example, without any blockers there are 4 combos of AKs (A♠️K♠️, A♥️K♥️, A♦️K♦️, A♣️K♣️). The fact you hold the Ah makes the 2nd of these impossible. And, the fact you hold the Ks makes the 1st of these impossible. Thus, leaving you with just 2 of the original 4 combos. You should note that if you held A♥️K♥️ (or any AK suited) in your hand, this only blocks 1 of 4 suited combos of AK suited.

##### Shortcut

A quick way of working out combinations, when you hold a blocker is to multiply the relevant unknowns together. If you are working out combinations of a pair, you must divide this answer by 2. This is because the order doesn’t matter.

AK

4 aces in deck x 4 kings in deck = 16 combos of AK

If you have the Ah and Ks in your hand, there are only

3 aces in deck x 3 kings in deck = 9 combos of AK

AA

(4 aces in deck x 3 aces in deck) / 2 = 6 combos of AA

If you have the Ah and Ks in your hand, there are only

(3 aces in deck x 2 aces in deck) / 2 = 3 combos of AA

He has QQ 42.9% of the time (i.e. 6/14) when you hold A♥️K♠️. This is 27.2% of the time (i.e. 6/22) when you hold JJ. Comparing your equity with A♥️K♠️ versus each part of his range (which you can weight according to their combos), and combining this information with pot odds, implied odds, SPR etc. again you can make a decision about how you will proceed in this hand.

## 3. Combinatorics Poker Maths

What’s the probability of getting, or your opponent holding, a particular hand or range? Use combinatorics, and probability to answer these poker maths questions.

• There are 1,326 different starting hands you could get (assuming the order of your two cards doesn’t matter). You need to have strategies on how to proceed for all of these.
• You could receive any of the 52 cards in the deck as your 1st card. Then, any of the remaining 51 for your 2nd card. When performing these calculations, it doesn’t matter how many other players have also receive cards. Whether, and how many of, the remaining cards other players have (or are still in the dealer’s hand) make no difference. You then multiply 52 x 51 = 2,652.
• There is one final step. As the order of the cards in your hand makes absolutely no difference, we divide by 2. Imagine the dealer gave you black aces. Whether you got the ace of clubs first and ace of spades second (or the other way around) doesn’t matter. So 2,652/2=1,326 unique starting hands.

If however you count all hands with cards of the same rank as equal (for example 2♥️2♠️ = 2♣️2♦️, or A♠️K♣️ = A♥️K♥️, then there are only 91 possibilities of hands a player can hold. 13 of these are pairs, and 78 of these are unpaired hands.

In many situations, ranges tend to be linear. This means if an opponent is playing a hand they are usually playing all better hands.

### You do NOT have a 1/x chance of each of the x possibilities of hands

#### Pair

This is because for any pair, say JJ there are 6 ways of receiving the pair:

• Jack of spades + Jack of diamonds/hearts/clubs
• Or, Jack of hearts + Jacks of diamonds/clubs
• Or, Jack of diamonds + Jack of clubs

#### Unpaired

For any unpaired hand, say KQ there are 16 ways of receiving that unpaired hand:

• K♠️Q♠️, K♠️Q♥️, K♠️Q♦️, K♠️Q♣️,K♥️Q♠️, K♥️Q♥️, K♥️Q♦️, K♥️Q♣️,K♦️Q♠️, K♦️Q♥️, K♦️Q♦️, K♦️Q♣️,K♣️Q♠️, K♣️Q♥️, K♣️Q♦️, K♣️Q♣️

As you can see out of these 16 ways, 4 ways are suited, and 12 are unsuited. This will be important in range analysis. Perhaps you think your opponent might be playing just the suited combos of say AK (or another hand). Or, all the combos, depending on the situation.

To check our poker maths, for the:

13 pairs, each has 6 combinations: 13 x 6 = 78 combos

78 unpaired hands, each has 16 combinations: 78 x 16 = 1,248 combos

Therefore, all hands (paired + unpaired) = 78 combos + 1,248 combos = 1,326 combos

You are in a position to work out the chance of receiving any specific hand, or range of hands using combinatorics and probability.

### Combinatorics and probability examples

#### Pocket pair preflop (if it doesn’t matter which one)

78 combos of pocket pairs divided by 1,326 total combos equals ~5.9%

You will expect to see a pocket pair dealt to you, ~5.9% of the time.

• There’s a beautiful way to get to this answer in a matter of seconds.
• Understand that to get a pocket pair, it doesn’t matter what card the dealer deals you first. All that matters is your second card matches the first card. How often will your second card match the first card dealt?
• There will be 3 remaining cards in the deck that match your first card (regardless of what it was). There’s 51 cards left in the deck. You do not have to consider cards our opponents receive, as we don’t know what they are.
• So, the answer is 3/51 =~5.9%.

#### A suited hand

There are 78 unpaired hand types, and each has 4 suited combos (spades, hearts, diamonds, clubs), e.g. AK has the following suited combos: A♠️K♠️, A♥️K♥️, A♦️K♦️, A♣️K♣️

So, there are 78 x 4 = 312 combos of suited hands

312 combos of suited hands / 1,326 total combos = ~23.5%

A quick way to work this out. It doesn’t matter what the first card the dealer deals you is. There are 12 remaining cards in the deck that are the same suit as your first card. This applies regardless of what the first card was. There’s 51 cards left in the deck. You do not have to consider cards our opponents receive, as we don’t know what they are. So, the answer is 12/51 =~23.5%

#### An unpaired unsuited hand

There are 78 unpaired hand types, and each has 12 unsuited combos

So, there are 78 x 12 = 936 combos of unsuited hands

936 combos of unsuited hands / 1,326 total combos = ~70.6%

A quick way to work this out. It doesn’t matter what the first card the dealer deals you is. There will be 36 remaining cards in the deck that are a different suit that also don’t pair your hand. This applies regardless of what the first card was. There’s 51 cards left in the deck. We do not have to consider cards our opponents receive, as we don’t know what they are. So the answer is 36/51 =~70.6%.

You can also work out the probability of receiving a particular hand:

#### AA

Each of the 4 aces, can be dealt with one of other three aces to make AA, thus there are (4 x 3)/2 = 6 combos of AA (we divide by 2 as order doesn’t matter, A♥️A♠️ is exactly the same as A♠️A♥️)

6 combos of AA / 1,326 total combos =~0.45%

This is the probability of receiving any specific pair, be it AA or 66.

#### AKs

There are 4 combos of any suited hand. Both cards must be spades, hearts, diamonds, or clubs.

4 combos of AKs / 1,326 total combos =~0.3%

This is the probability of receiving any given suited hand, be it AKs or 74s

#### JTo

Each of the 4 jacks, plus one of three tens, makes JTo. The fourth ten will be the same suit as the jack in question. Thus there are 4 x 3 = 12 combos of JTo

12 combos of JTo / 1,326 total combos =~0.9%

This is the probability of receiving any given offsuit hand, be it JTo or 92o

#### KQ (either offsuit, or suited)

Each of the 4 kings, plus one of four queens, makes KQ. Thus there are 4 x 4 = 16 combos of KQ.

16 combos of KQ / 1,326 total combos =~1.2%

This is the probability of receiving any given unpaired hand, be it KQ or 72

## 4. Minimum defence frequency Poker Maths

Minimum defence frequency (abbreviated as MDF) is the percentage of time a given player must not fold in response to an aggressive action from their opponent (either a bet, or a raise) to prevent their opponent from taking that aggressive action with any two cards.

### Calculating minimum defence frequency

Let us look at an example:

In a 9 handed \$1/3 live cash game, a player in middle position opens for \$12. We call on the button. The blinds fold. Thus, there is \$28 in the pot, going into the flop. Both players check the flop and the turn. On the river the middle position player decides to bet \$14 (i.e. a half a pot sized bet). What is our minimum defence frequency?

The best way to to do the poker maths is using 2 steps:

1. How often does the villain need us to fold, for his bet to break even?
The villain is risking \$14 to win the \$28 already in the pot. This means he is putting in \$14/\$42 = 33.33% of the new total pot size after his bet. So he needs us to fold 33.33% (or 1/3) of the time, in order to break even.
2. Given this how often do we need to defend to stop him breaking even?
He needs us to fold 33.33% (1/3) of the time, in order to break even. So if we defend more than 66.67% (2/3) of the time, he does not break even.

### Poker Maths formula for minimum defence frequency

MDF = 100% – (Villain’s break even %)

OR

MDF = 1 – (Villain’s break even fraction)

### Don’t confuse pot odds & MDF

In the above example, when villain bets \$14 into \$28 on the river, we are getting 3:1 pot odds. Call \$14 to win a new pot size of \$42? This means we only need to win 25% of the time when we call. This means we should be looking to call with all hands that have at least 25% equity, versus the range that we put villain on.

If we don’t have enough hands that have at least 25% equity versus villain, such that we can call 33.33% of the time, villain can bet with any 2 cards and make a profit.

### Common MDFs (to avoid having to do any Poker Maths)

• When facing a:
• 1/3 pot sized bet, your MDF is 75%
• 1/2 pot sized bet, your MDF is 67%
• 2/3 pot sized bet, your MDF is 60%
• 3/4 pot sized bet, your MDF is 57%
• 1/1 pot sized bet, your MDF is 50%
• 2/1 pot sized bet, your MDF is 33%

## 5. Pot Equity Poker Maths

Pot equity refers to the odds that a particular hand (or range of hands) will win, given the hand(s)/range(s) held by opponent(s), AFTER the river card is dealt.

Equity is usually expressed as a percentage, but can also be expressed as a ratio. It is important to note that the equity of a hand will usually go up or down as the hand progresses. If you have ever watched poker on television, and see a percentage shown next to a player’s hole cards. This indicates the player’s current equity in the hand. Quickly combining equity calculations, with pot odds calculations, is an important skill to have.

### Example

Let’s say we are playing \$2/\$5, and it folds to the cutoff who is holding KdKc. The cutoff (who has \$500 in his stack) opens to \$20. The button (who covers the cut-off), who has AhAs, 3-bets to \$60. It folds to the cut-off, who 4-bets to \$200. The button goes all-in for \$500, and the cut-off quickly calls.

Just before the flop, the CO with KdKc has 18.74% equity. The button with AhAs has around 81.26% equity.

The exact suits each player is holding can impact these percentages.

If any other player has folded an A or K, this does NOT matter as we cannot know they have done this. You should note if you watch poker on television, the equities shown usually take account of the cards other players fold, where they are known.

If however, one or more cards are accidentally exposed by the dealer or another player (which does happen from time to time in live poker), this will have to be taken into consideration in your equity calculations.

#### If the flop comes Ks Qd Js, equities change considerably.

The CO’s KdKc has 75.96% equity, whilst the button’s AhAs has 24.04% equity.

CO has a set, but button is looking to hit one of the 4 remaining tens in the deck (which would make him a straight), or one of the 2 remaining aces in the deck (for a higher set than his opponent). That’s 6 direct outs to improve.

Of course, even if button makes a straight on the turn, cutoff can still win the hand if the board pairs on the river (giving him a full house). If the button makes either a straight or higher set on the turn, cutoff can still win if the river is the last king in the deck (the case king) which will give him quads. That is to say, your equity is diluted.

Button also has backdoor outs (and thus, backdoor equity in this hand). Backdoor equity/outs means that he will need 2 running cards. Both the turn and river need to be useful if he is to win the hand. In this example, two more spades would win button the hand. This is as long as the board does not pair, which would give CO a full house.

### Equity types

• Hand v Hand
• You can easily memorise estimates for Hand v Hand equity
• For example, an overpair usually has around 80% equity, versus an underpair
• Two overcards usually have around 45% equity, versus an underpair
• Hand v Range
• i.e. What is the equity of your known hand (e.g AhKs) vs the range of hands you have assigned your opponent (e.g AA, KK, QQ, AK)?
• Range v Range
• i.e. What is the equity of the range of hands you would normally hold following a sequence of actions, versus the range of hands you put you opponent on?

You will usually be thinking either in terms of hand v range equity, or range v range equity. You can calculate Hand v Range and Range v Range equity instantly from the table using an equity calculator. There are many good pieces of software, available on a variety of platforms, that are totally free to use. The best way to get a feel for equity calculations is to perform these regularly using software. You will start to notice patterns or get a feel for the likely equities in a variety of situations. Then, you can estimate these accurately at the table when a similar spot comes up.

### Hot and cold equity

When we say that AA has around 80% equity versus KK which has around 20% equity, we are talking about ‘hot and cold’ equity.

Hands with the most hot and cold equity, are not necessarily the best hands to be playing. Clearly A7o has more equity then KQs. Imagine you could get all the money in with A7o knowing your opponent had KQs, in a heads up scenario preflop. Clearly you would want to do so. In real life, you are unlikely to know what hand your opponent has. Instead you can only put them on a range of possible hands. In a 9 handed poker game, with deep stacks, you will almost certainly be folding A7o preflop. This is unless it is unopened to you on the button or you get to check your bb. Or, you are getting a good price to continue in the bb). You will however likely be at least thinking about playing KQs from all positions.

### Retained equity

We prefer hands that will retain their equity versus the hands our opponents ranges for continuing in the hand (whether they do so by raising, betting, or calling). Being ahead in equity terms of hands our opponent will likely be folding before showdown (be it preflop, on the flop, turn, or river), doesn’t help us.

Imagine a strong studied player opens in early position in a deep stacked live \$2/5 game. It is folded to us on the button, would we rather have A9o or 87s? Clearly, A9o will have more hot and cold equity than 87s. This is versus any reasonable range of hands we put our opponent on. However, A9o is almost certainly fold in this situation. However, we will always be continuing (whether by calling, or 3-betting) with 87s.

### Outs

In the above example we talked about outs. Do you want to calculate your equity, when you think you are behind after any of the flop/turn/river have been dealt? It can be useful to think in terms of outs. Once you know how many outs you have it is easy to estimate your equity as we will show below. Outs are the number of cards remaining (i.e. that you know could receive) that you think will give you the winning hand.

When you have a draw, and you think you will will have to hit your draw to improve, you will want to think about your outs

#### If you have a flush draw, you have 9 outs

For example if you hold KdTd and the flop is 9d5c2d, the following 9 cards will give you your flush:

Ad, Qd, Jd, 8d, 7d, 5d, 6d, 4d, 3d.

It is important to note you may have other outs too. Will making a pair of kings might be enough to win the hand? The,n perhaps you have 3 further outs (Ks, Kh, Kc).

You may also have backdoor outs (such as hitting a running straight, or running two pair, or trips). Note that, even if you hit your flush draw it might not be enough to win the hand. Imagine if your opponent held 99/55/22 on this flop. If the board pairs on the turn or river, you cannot win this hand unless you hit a running royal flush.

#### If you have a straight draw, you have 4 or 8 outs

• A gutshot (also known as an inside straight draw) has 4 outs.
• An open ended straight draw has 8 outs.
• A double belly buster straight draw has 8 outs. For example, imagine the flop is Tc9d6s, and you are playing Qd8d. You will make a straight with any J or 7.

#### Not sure a card is an out?

Let’s say you are mainly going for a flush, as the example immediately above. However, you think making a pair of kings might also be enough to win the hand. However, you are not sure. How about counting either 2 or 1 (instead of all 3 remaining kings) as outs?

### Converting outs to equity quickly (to simply poker maths required)

#### Rule of 2

After the turn (i.e. there is 1 card to come), multiply your outs by 2. That is to say count your outs, and double it. This gives you a fairly accurate estimate of your equity, expressed as a percentage.

The reason this works is that there are 46 unknown cards on the river. That’s 52 cards in the decks – 2 in your hand – 3 on the flop – 1 on the turn. If it was 50 unknown cards to get a % you would multiply by 2. With 46 unknown cards, you get a reasonable estimate, by multiplying by 2 (but clearly there is a small error).

#### Rule of 4

After the flop (i.e. there are 2 cards to come), multiply your outs by 4. That is to say count your outs, and double it twice. This gives you a fairly accurate estimate of your equity, expressed as a percentage.

You have probably deduced why this works, after reading the above explanation. As you have 2 chances to hit your outs (i.e. both turn, and river), your equity approximately doubles. Again, this simplified rule has a degree of error. There are 47 unknown cards on the turn, and 46 on the river. Yet, this formula is assuming there are 50 unknown cards on both the turn and river.

#### Improving the rule of 2 or 4

We have shown you why the rule of 2 and rule of 4 produce good but not perfect estimates. To reduce absolute errors, we would suggest the following modifications, which are certainly not onerous to do at the table.

• You can actually improve on the rule of 2, with almost no additional effort.
• Just add 10% to your answer.
• So, if you had 7 outs, use the rule of 2 formula to estimate 14% equity.
• Then you add 1.4% to this answer, giving you 15.4% equity.
• The rule of 4 is actually pretty accurate, until you have a larger number of outs.
• So we suggest you use the rule of 4 as is for 1-9 outs.
• If you have 10 or more out we suggest multiplying your outs by 3, and then adding 9.
• So, if you had 15 outs, you would do the following calculation in your head: 15×3=45%, and then add 9%, giving you 54% equity.

By taking a few more seconds to do this, you will reduce your absolute errors in your equity calculations. The more accurate your equity estimates, the better your decisions might be. Using these modified rules (which are still easy to do), means your equity calculations will a low maximum absolute error. This will be around 1% (provided you have accurately counted your outs).

## 6. Expected Value Poker Maths

The expected value of a cash game poker hand is the amount of money we expect to win, on average, for a given line. All money previously invested into the pot, is treated as dead money.

• We do not have to deduct our previous investments into the pot, when calculating expected value.
• Another term for Expected value is expectation. Also, the abbreviation EV.
• Expected Value in cash game poker will be a \$ (or £,€, or any other currency) amount.
• In tournament poker, you can also look at chipEV. This is the amount of chips we expect to win, on average, for a given line.

### Example

• Let us imagine after all the turn betting, we and our only remaining opponent are all-in.
• There is \$600 in the pot. We put in \$290 of this, our only remaining opponent put in \$290. \$20 is from players who have folded.
• Let’s say in this all-in scenario both players have decided to show their cards. You calculate your hand has 90% equity, versus your opponent’s hand. This means your expected value is \$600 x 0.90 = \$540.
• Notice, we do NOT deduct the \$290 we have invested, from the expected value calculation as this is dead money. Once we have put it into the pot, it is no longer ours.
• Of course our \$290 we have put in (as well as all the other money any other opponent has put in) is considered in the pot size, in the expected value calculation.

#### No further money

The ONLY time expected value can be calculated as (total pot size x equity) is IF all players remaining in the hand cannot place any more money into the pot. If further money can be placed into the pot, the total pot size is not yet known. So, this is not the way to calculate expected value.

### EV is different from equity

Whilst equity and expected value are correlated, they should never be treated as the same concept. The equity of hand are the odds (usually expressed as a %, or ratio) of a particular hand/range to be the best hand when the river card has been dealt.

### The expected value of folding is 0

The definition of expected value states that all money previously invested into the pot is dead money. Thus, it follows that if a players fold their hand at any point their expected value is zero. You do not have to put in any more money to fold. And, anything previously invested (even on the same street) is not relevant.

### Results & EV differ in the short term

In any given poker hand (except in the case of a tie) you will usually win the entire pot. Or, end up with absolutely nothing (and thus lose everything you have invested). Let’s say in a cooler situation, you and your opponent are all-in before the flop. There is \$1,000 in the pot, consisting of the money you and your remaining opponent have put in, plus anything from any players that have folded. One of you has AA and one of has KK. AA has an EV of approximately \$1,000 x 0.80 = \$800. KK has an EV of approximately \$1,000 x 0.20 = \$200. However, in this one hand one of you will win \$1,000, and one of you will win \$0. 20 times out of 100, it will be the AA who will win \$0.

Expected value is a long term concept. It tells you how much you will win, on average, if this hand was played from this point a large number of times.

### Max your EV = goal in cash games

Short terms results in poker will be affected by the variance that is built into the game. As a studied poker player, variance is one of the things that is helping you take the money from the unstudied or weaker players, over the long term. If there was no variance in the game, the most skilled players would win the money of the less skilled players faster, and unskilled players would quit at a faster rate. Variance means unstudied players may be rewarded in the short term for what are long term poor strategies, which may mean taking the money from studied players playing better long term strategies. However this is exactly what you want, and what makes poker so profitable. As a studied player, your goal should always be to maximise your long term expected value.

### Dynamic fold equity EV

One of the most complicated looking (but not hard to follow), EV calculations is when you are calculating Dynamic Fold Equity. This is when you are considering a semi-bluff, but are wondering if it is worthwhile.

#### A semi-bluff is when you hope to win the pot in 2 ways

• (a) either your opponent folds and you win a smaller pot, or
• (b) your opponent doesn’t fold but you make a better hand than your opponent, and win a bigger pot.
• Of course, sometimes your opponent won’t fold, but you also don’t make a better hand than your opponent. You will need to take this into account.

An example of a semi-bluff is if your suited connector after the flop has a flush draw. 2 more cards of your suit arrive on the flop, which didn’t give you a pair or straight draw. You believe you are up against a weak top pair hand (i.e. top pair, weak kicker). You believe you will always win the hand if another club comes. But, won’t otherwise win except for a small amount of backdoor equity. You were the preflop raiser, and your opponent (in the big blind) who called preflop checks to you. To decide whether to proceed you will have to calculate your EV.

#### You need to consider all of the following

• A = If your opponent folds, how much will you win?
• (frequency you expect your opponent to fold x pot size without counting our uncalled semi-bluff)
• B = If your opponent calls, and you make your flush (or otherwise win the hand via backdoors), how much will you win?
• (frequency you expect your opponent to call) x (hero’s equity) x (pot size before our semi-bluff + the amount of our semi-bluff)
• C = If your opponent calls, and you don’t make your flush (or otherwise win the hand via backdoors), how much will you will lose?
• (frequency you expect your opponent to call) x (opponent’s equity) x (the amount of our semi-bluff)
• EV = A + B – C

## 7. SPR Poker Maths

Between any 2 players the stack to pot ratio (abbreviated as SPR) is the smallest stack size remaining in front of either player (known as the ‘effective stack’ between them), divided by the current pot size. Note that the SPR is calculated after a round of betting has been fully completed. This can be preflop, on the flop, turn, or river.

The higher the stack to pot ratio, the more money or chips behind left that can be played for.

### Example of calculating stack to pot ratio

• Live \$2/\$5, 9 handed cash game.
• The UTG player sitting with \$1000 opens to \$20. It folds to the short stacked button who is sitting with \$161. The button calls, and both the blinds fold.
• There is \$47 in pot just before the flop. That’s \$20 from UTG, \$20 from the button, \$2 from the small blind, and \$5 from the big blind. The UTG has \$980 (\$1,000-\$20), and the button has \$141 (\$161-\$20) remaining. The effective stack going into the flop is thus \$14. The effective stack between any two players, is the smallest of their remaining stacks. The extra chips the UTG player has, are not usable.
• The SPR is Effective stack (\$141) DIVIDED BY the Pot size (\$47) = 3

### Easy to remember SPR tactics

You can use the Stack to pot ratio, to help decide whether to stack off with top pair or better. Or, even worse hands in certain situations. This is especially useful when there is little other information. Put another way, SPR can help you decide about your commitment to the pot. This is regardless of what happens on future streets.

Tournament players will need to consider all their incentives at any give time, of which SPR will be just one.

• Warning: The following are really rough tactics, for when you have top pair or better. You should NOT blindly follow these – always consider all the other information at your disposal and your overall strategy:
• If you have at least top pair, stacking off with an SPR of at least 6 may be negative EV.
• If you have at least top pair, stacking off with an SPR of 3 or less may be positive EV.
• When your SPR is between these amounts, then whether you should stack off might be based upon different factors. For example board texture, and if playing exploitatively what reads you have on your opponent.

A big mistake to avoid is to getting too attached to your preflop premium hands in high SPR pots.

Another big mistake to avoid is folding your top pair+, in low SPR situations. Your hand might be winning, and if it isn’t you still have equity versus their opponent(s) range.

#### Different preflop starting hand types prefer different SPRs

• Speculative/drawing hands like small pairs (aiming to make sets)prefer the SPR to be high. The same goes for suited connectors (aiming to make flushes, and straights). This is because drawing hands would ideally like opponents to either fold. Or, win a bigger pot when their draw comes in. The smaller the SPR, the harder it is to get opponents to fold. Thus, the less you could win if your draw comes in. If the SPR is too small, you might not have the correct odds (neither direct or implied) to proceed in the hand when facing aggressive action from your opponent.
• Premium pairs and premium no-pair starting hands (such as AKo) would prefer a low SPR. The lower the SPR, the easier it will be for you to realise all your preflop hot and cold equity. This is since an aggressive opponent will find it harder to get you to fold.

#### Betting decisions & stack to pot ratio

• If your hand/range would prefer a lower or higher SPR, you can use your opening sizes and re-raising sizes to help get to the SPR you want.
• Whether you fastplay or slowly, and your line in a hand, might depend on SPR.
• If you are the out of position preflop raiser in a heads up pot postflop, and the SPR is on the low side and you have a draw – you could consider checking, and then over bet shoving the flop. Whilst with the same flop and draw in a bigger SPR, you might c-bet yourself.