Poker Math Reference e-Book.

Table of Contents.

1. Basic Combinatorics in Poker (pg.1)2. Calculating a Pot Sized Raise (pg.2)3. Risk and Reward (pg.3)4. Preflop Math (pg.4)5. Blind Stealing (pg.4)6. 3 Betting (pg.5)7. 4 Betting (pg.5)8. 5 Betting/Working out a 4 Bet Range (includes some range building) (pg.6,7)9. Calling a 3 Bet Shove (pg.8)10. Calling a 4 Bet Shove (pg.9)11. Calling a 5 Bet Shove (pg.10)12. The Big Poker Formula (pg.11)13. 5 Bet Shoving Light (pg.12)14. 4 Betting with the intention of calling the 5 bet shove (pg.14)15. Semi-Bluff Shoving the Flop or Turn (pg.15)16. Bluff Betting and Bluff Raising the River (pg.18)17. Calling the River Bet. (pg.20)18. Thin Value Betting the River (pg.21)

EV Calculation Spreadsheet included.Most of the calculations in this reference book, deal with stealing or bluffing. When we're betting forvalue all we really need to find is more combinations in our opponents range which are worse than ourhand. So, there is not as much need for as much math to see if we're making +EV bets.

BASIC COMBINATORICS IN POKER (pg1)

Combinatorics in poker is basically to do with working out how many combinations of hands there are in certain situations.For example :

• How many ways we can be dealt AA• How many ways we can be dealt AK• How many combinations of AK are there on a A32• How many combinations of AK on a AK3• How many combinations of open ended straight draws on a KT7

AA combos = AsAc, AsAd, AsAh, AcAd, AcAh, AdAh 6 total combos.AK combos = 4 suited AsKs, AcKc, AdKd, AhKh,12 non suited – AsKc, AsKd, AsKh,AcKs, AcKd, AcKh,AdKs, AdKc, AdKh,AhKs, AhKc, AhKd. 16 total combos (4 Aces X 4 Kings =16)

After we have seen a flop, there will be some card removal. e.g.

Flop Ah3s2c. Now there is one Ace on board, so there is only 3 other Aces and 4 other Kingsthat could have potentially been dealt to our opponent. 4X3=12. So only 12 potential AKcombos could have been dealt to our opponent.

AhKs3d

In this instance there are only another 3 Kings and 3 Aces, 3X3=9 combos.

Combos of open ended straight draws on a KT7?Well 98 and QJ both give an opponent an open ended straight draw. 16 combos of each. So our opponent could potentially have 32 combos of open ended straight draws in his range of hands.

Online poker players can use poker software such as Hold Em Manager to help find anopponents total range, as well as using some hand reading skills on each street. This referencebook is for people who have already acquired this skill.

I want this book to just be used for all the different math situations that occur while playing ahand of poker. There is a spreadsheet included for every single poker calculation included inthis reference book. Starting off with the following simple method of calculating a pot sizedraise -

CALCULATING A POT SIZE RAISE. (pg2)

Formula 1 on the spreadsheet.

A lot of poker sites will have a raise pot button, some of them don't. If you're playing live it's probably good idea to know how to calculate a pot size raise. So this will help the online players who play on a site that don't have the raise pot button to help them out, and live players. The spreadsheet can be used to quickly calculate a pot size raise. All you need to do is input the opponents bet size that we're facing and the size of the pot after our opponent bets.

e.g. Opponent Bets 10 big blinds into a 10 big blind pot.Opponents Bet Size = 10bbsPot after opponent bets = 20bbsPot Size Raise = 40bbs.

The new pot is now 60bbs and our opponent has to call 30bbs to win 60bbs giving him pot odds of 2/1.

To calculate it mentally. Double the opponents bet and add it to the pot after he has bet.In this case the opponents bet was 10. We double that = 20The pot after he bets = 20 20+20 = 40 pot size raise.

One more example. Opponent bets 3 into 6 a 1/2pot bet.Double the bet of 3 and add it to the new pot of 9. = 15 bbs. We raise to 15 the new pot is 24and our opponent has to call 12. 2/1

RISK AND REWARD (pg3)

There is a risk versus a reward in nearly every poker decision or bet or call that we make at the table. Here is a table that will be referenced throughout this booklet. In the bluffing column, we can see that if we make a pot raise, we need our bluff to be good, or work 67% of the time. If we bet the pot as a bluff it needs to work 50% of the time. We risk a pot bet to win the pot. X to win X. All the percentages are there for most standard bluff bet sizes.

When we have to call the percentages are not as severe as there is usually more money in the pot. So, if our opponent bets the pot, when we call, we only have to be good 33% of the time. We will be getting 2/1 on our call. These type of figures are great if we are facing a bet on the river and there is no more action to come. In the following chapters you will see more examples of where we can look at bet sizing and then figure out how often we need to be good in certain spots to make our call or bet profitable. After that we need to look at our equity vs our opponents and then we can get a good handle on if we are making the correct plays.

(this table was originally shown in the book - “Poker Math That Matters – By Owen Gaines”)

PREFLOP MATH (pg4)

Blind Stealing.

Spreadsheet formula 2.

When ever we steal preflop we obviously make a bet. We risk a certain amount of money with our bet to win the blinds.

In the small blind there is 0.5 blinds and in the big blind there is 1 big blind. Usually we will raise around 3 big blinds to try win these blinds.

So we risk 3 big blinds to win 0.5 + 1 big blind. These numbers can be inputted into formula 2 to see how often we need this bet to work for it to be profitable.

Here's how the formula works on paper.

We risk x to win x + y. x/(x+y)x = our pre flop raise sizey = the blinds in total.

3/(3+1+0.5) - 3 / 4.5 = 0.667 = 67%

We need this bet to work 67% of the time for this bet to profitable. Online players familiar with their Heads Up Display (HUD), will be able to access a stat which tells us how often players fold in the blinds versus a blind steal. If we see a Fold to steal stat of 80% by the blind players, we can start to steal super wide.

Once we make this bet or steal raise there's always a chance that the blinds will try to win the pot themselves with a reraise or 3 bet. (the big blind is the first bet, the steal raise is the 2nd bet, and this reraise is the 3rd bet. Also known as a resteal).

3 Betting or Restealing. (pg 5)

We use the same formula 2.

So, let's say now we're in the blinds and we're facing a steal raise from the button. We can see from our HUD stats that he steals from the button 40% of the time. He also folds to a 3bet 70% of the time vs a 3bet from the blinds when he is on the button. All these specific positional stats can now be found using Hold Em Manager.

We're in the big blind and we decide to 3 bet to 10. The actual bet is 9 as we have already paid 1 big blind from being in the big blind.

We can still use the same risk reward formula to see if our bet will be profitable.

We risk 9 to win the small blind (0.5), the big blind(1) and the original steal raise(3).

x/(x+y)

9/(9+0.5+1+3) - 9/13.5 = 0.667 = 67%

We know from the HUD stats that he usually folds to a 3bet 70% of the time in this spot. Making this bet immediately profitable.

4 Betting Preflop

Let's switch the roles again, we did our steal raise and our opponent 3bet us from the big blind. We look at our HUD stats and see that our opponent 3bets a button steal from the blinds 12% of the time, we know that this is a wide range of hands that our opponent is stealing with. We check to see what his fold to 4 bet percentage is and it is 75%. We decide that a 4 bet in this spot could be profitable, a fairly standard 4 bet size in this spot would be to 25 big blinds.

Once again we go to the risk/reward formula.

We risk 22 to win the 3bet(9), the steal raise(3), the small blind(0.5) the big blind(1)

22 / (22+9+3+0.5+1) = 0.62 = 62% of the time this bet needs to work to be profitable.

We know from the opponents HUD stats that he folds 75% usually in this spot. Making this another profitable bluff spot.We would need quite a lot of hands on players for this 4 bet fold HUD stat to be accurate.

5 Betting and Working out our Opponents 4 Bet Range. (pg6)

Before we make our 5 bet bluff, it's important to know our opponents 4 bet range.Our opponent preflop raises from the button 40%. We have 3 bet him, now we're facing a4 bet from our aggressive opponent.

We can use our hud stats to figure out a 4 bet range for our opponent.

We can also use the call 3bet% and the fold to 3bet% stat as well as the 4bet% to get a pretty good handle on our opponents ranges. It's important we have a 1k+ hands on our opponent to do this kind of thing, and use Hold em manager to see some hands that our opponent has called a 3bet with and 4 bet with.

eg. Our opponent -Folds to a 3bet 60%.Calls a 3bet 20%4Bets 20%

His original button range was 40%.

Now we can find these individual ranges-

Fold 60% = 0.6 * 40 = 24% (bottom 24% of original 40% range)Call 20% = 0.2* 40 = 8% (8% under top of original 40% opening range)4 Bet 20% = 0.2 * 40 = 8% (the top 8% of hands of the original 40% opening range)

His 4bet range in this spot is the top 8% of his wide preflop raising range, with a few bluffs mixed in more than likely.

An example of this range:-

So now if we want to 5bet bluff we need to figure out how often he will call a 5 bet from this range. Let's say he calls a 5bet pretty tight and only calls with the top hands in his range, eg. AK,QQ,KK,AA.

These are the top 2.6% of hands in his 8% 4bet range.The stacks are sized so that he can only call or fold to the 5 bet.So he calls - 2.6% of his original 8% 4bet range.

2.6/ 8 = 0.325 = 32.5% of the time he calls.

So if he calls 2.6 - he folds 8-2.6= 5.4 5.4/8= 0.675 = 67.5% (67.5+32.5=100)

Call % = 32.5% Fold % = 67.5%.

So now, let's get back to our poker situation. Our opponent steal raised on the button 3bbs, we 3bet from the big blind to 10, our opponent 4bets 22bbs from 3 to 25bbs, and now we shove all in 90bbs up to 100bbs.

We can still use the risk/reward formula 2 to see if this bet size will be profitable.90 / (90 + 25 + 10 + 0.5) =0.72 = 72%.

This bet on it's own is not immediately profitable as we need our opponent to fold 72% and we can see by our other calculations that he will only fold 67.5%. But what this equation doesn't take into account is the fact that when our opponent calls we will still sometimes win this hand even when we're bluffing. Most of the calculations in the rest of this book will start to take into account the equities and outcomes when our hands clash and we reach a showdown.

The EV of Calling a 3 Bet Shove Preflop. (pg8)

This will usually occur vs a short stacker. In these instances we need to find our equity vs our opponents shoving range and then use formula 3 to find out if we have the correct odds or to make the call. These calculations can be useful for MTT's and STT's as well as vs short stacking cash game players.

eg. We raise to 3bbs Ace Ten Suited Hearts on the Button in a 100 nl cash gameThe small blind folds but the short stacker in the big blind shoves for 30bbs.Can we profitably call here?

First we need some info on our opponents shoving range. Let's assume he is a fairly solid short stacker and we have 2k hands on him and we quickly check our notes to see that we have seen him shove pretty wide vs the button and he has shown down 66,88,JJ,AA / ATs,AKs and AJo + KQo. So we give him the range of 66+,ATs+ and KQo, AJo.Now we can use an equity calculator such as Poker Stove to calculate our equity vs this range.

Hero EQ = 40%Villains EQ = 60%

We have to call 27bbs to win -Our opponents shove = 30bbsOur Pre Flop Raise = 3bbsThe Small Blind = 0.5bbs (33.5bbs)

We can use formula 3 on the spreadsheet to find our break even call%.Cost of our call = 27bbs.Pot before our call = 30bbs+3bbs+0.5bbs(33.5bbs)=0.44% We need 44% equity to call this.

This is essentially another risk/reward calculation. X/X+YWhere X is the cost of our call and Y is the Money in the pot before we call.27/(27+33.5)=0.446

Another way to work this out is to use formula 4 to find the actual EV. And this is a good way to check to see if the first calculation works out.

Using the Formula - EV = (-LV+WH)EV = Expected ValueL = Maximum Loss or Cost of Call (-27) (on the spreadsheet just type 27 not -27)V = Villains Equity (60% or 0.6)W = Maximum Win (33.5bbs)H = Hero's Equity (40% or 0.4) (on the spreadsheet if we input the hero's equity the villains equity will automatically be worked out.)

EV= (-27*0.6)+(33.5*0.4) = -2.8bbs EV

Using the same formula 4, if we input 0.44 (0.44628) into the H (Hero Equity) column we can see this brings the EV to very close to zero, which as we calculated using the previous formula should be the break even call %.

(pg.9)The following examples of calculating the EV of calling a 4bet or 5bet shove are pretty much identical to this. We just have to be careful when we are figuring out the maximum win and maximum loss amounts.

Calling a 4 Bet Shove.

Example.The Laggy Cut off with a 50bb stack Raises to 3bbs. We 3 bet with AsQh on the

button to 11bbs. Both of the blinds fold and our opponent in the cut off 4bet shoves 47bbs. His opening cut off range is 20% and he has a 4bet of 25% .

First we find his shove range. -0.25*20 = 5% He shoves the top 5% of his range. (Ats+/AJo+/KQs/TT+)

We will be using the same 2 formulas. Formula 3 and 4.

Hero's Equity. - 45%Villains Equity – 55%

Cost of Our Call – 39bbsPot after our opponent Bets – 50(money opponent has put in the pot) + 11(our 3bet+1.5(blinds)

Pot = 62.5bbs.

Risk Reward x/(x+y) 39/(39+62.5) = 0.384This is the break even call %. (38.4%)

Now we go to formula 4. EV = (-LV)+(WH)

-L = -39 (type in 39 on spreadsheet the sum on the spreadsheet will already make the number negative)V = 0.55W = 62.5 (50+11+1.5)H = 0.45

EV = (-39*0.55)+(62.5*0.45)

EV= + 6.675bbs (You can also use monetary values in the M and L columns and the result will calculated in dollars or whichever monetary value you have used).

If we change the Hero column to 0.384 we can see that the EV changes to close to zero, comfirming our break even call % in formula 3 was correct.

Calling a 5 Bet Shove (pg10)

Example.

We raise first in on the button 100bbs deep with JhJc to 3bbs. Small Blind (100bbs dp)3bets to 11. Big Blind Folds and we 4bet to 25bbs. Our laggy opponent in the SB shoves for 100bbs.

We have 3k hands on our opponent and have seen him show up here with TT+ and AQo+.

Hero's Equity – 47%Villains Equity – 53%

Same formula.

Cost of Call = 75bbs.Pot after opponent shoves = 100 + 25 (4bet) + 1 (big blind) = 126

Break Even Call % = 0.37 (37%)

EV = (-L*V)+(W*H)

EV = (-75*0.53) + (126*0.47) = +19.47 EV.

Once again if we change the Hero column to 0.37313 we see the EV go close to zero.

The Big Poker Formula. (formula 5) (pg11)

EV = (F*P) + (1-F)*(-L*V)+(W*H)

We will be using this formula to calculate some slightly more complicated scenarios, and once you understand how this equation works then all other poker calculations seem super simple. This formula is relatively simple too, once you break it down and understand how it all works. This equation is used when our opponent has an option of folding or calling.

The First part of the equation helps us to calculate what we stand to win when our opponent folds.(F*P)

And the middle part gives us the call % (1-F).

The end part of the equation which we have already used deals with figuring out how much we stand to win or lose when our opp calls our bet or raise.(-L*V)+(W*H).

Once we figure out the value for F (fold equity), the 1-F is just giving us the frequency of ouropponents call.

EV = Expected Value---F = Fold Equity (How often we expect our opponent to fold to our bet or raise)P = Money in the Pot Before we Bet or Raise or Money we win the times when our opp folds.L = Maximum LossV = Villains EquityW = Maximum WinH = Hero's Equity.

This calculation works best when after our opponent folds or calls to our bet or raise there areno more possible betting rounds.

Eg when we shove the flop, turn, or when we shove all in preflop.

Once again it mainly deals with when we're doing this as a bluff or a semi bluff. If we're shoving for value all we need to do is find enough combos in our opponents range that are worse than our hand.

EV of 5 Bet Shoving Light Preflop. (pg12)

This calculation also works for 3 Bet shoving or 4 Bet shoving preflop. The calculationsfor figuring out the maximum loss and maximum win are much simpler than the 5bet shove. So I will show how to calculate the 5bet shove and I think figuring out how to use this calculation for 3bet shoving or 4 bet shoving light should be pretty self explanatory.

Example.Our opponent (100bbs deep) steals on the button 3bbs with a range of 40%. We

are in the Big Blind with 100bbs and we 3bet Ah2h to 10bbs. Our aggressive opponent 4bets to 25bbs, we have 4k hands on our opponent and his stats say he is 4betting 25% on the button and so far we have only seen him call a 5bet with QQ+/AK. With this in mind we decide that with our Ace blocker this is a good spot to 5 bet bluff shove. (We have an Ace which means he only has 3 combos of AA possible in his range).

So let's take a look at this. There's lots of information we need to find out for this to make sense. Let's break it down. This time I will figure things out in the order of the equation.

EV = (F*P) + (1-F)*(-L*V)+(W*H)

F = Fold equity.

Our opponent has raised on the button with a 40% range, and now he 4bets us with a 25% stat showing over a large sample size.

0.25*40=10% So our opponent has a fairly wide 4bet range of 10%. (please be warywith the HM2/PT4 stats, we need a lot of hands or samples to trust these stats.)

His 5bet call range is QQ+/AK Without taking into account Ace blockers and other factors, I'm trying to keep things simple to make the math easier to understand. But remember that the Ace blocker skews some of the math in our favour a touch.

His 5bet call range is 2.5% (2.6% to be precise QQ+AK)His 4bet range was 10%

So our Fold equity is 10-2.5 = 0.75 = 75%F = 0.75

------------------P= Money in the pot we win if we get our opponent to fold.

Let's work this out.

Opp pfr to 3bbsSB folds 0.5bbs

(pg.13)We 3Bet to 10bbs (9bbs +1bb posted)Opp 4bets to 25bbs.

25bbs total from opponent + 0.5 SB + 10bbs from us. = 35.5bbs total.

P=35.5bbs

That's the first part of the equation figured out.

(F*P) (Fold Equity * Money In Pot) (F*P) = (0.75*35.5)

Part 2 (1-F)This part is so we can work out our call equity.

1-0.75 = 0.25 (if he is folding 0.75 he has to be calling the other 0.25)

Easy enough.

Part 3.

This is how the equity will work out when the hands clash.

(-LV)*(WH)

L = Maximum Loss = In this instance,this is how much we shove. We shove 90bbs. (Our 3bet was 10bbs)V = We go to poker stove We have A2s hearts our opponent has JJ/AK 70.5 EQ for villain.W = Maximum win 100bbs from our opponent + My 10bbs + SB = 110.5)H= 29.5% Equity.

Now we have all the numbers. The more you practice these equations the easier it will become.Also, the assumptions about peoples ranges will depend on your experience at the table. I just hope I can help you figure out the math. Anyway, Let's plug in the numbers.

F = 0.75P = 35.5bbs1-F = 0.25-L = 90bbsV = 0.705%W= 110.5bbsH= 0.295

Plug these numbers into Formula 5!

EV = (F*P) + (1-F)*(-L*V)+(W*H) (pg.14) (0.75*35.5)+(1-0.75)*(-90*0.705)+(110.5*0.295)=

(FP) = 26.6(1-F) = 0.25(-LV) = -63.45(WH) = 32.59

(26.6) + (0.25) * (-63.45)+(32.59) = +18.66bbs

Now the Math for 4betting with the Intention of calling the 5Bet Preflop.

Example.We 4bet JJ and we decide to call our opponents 5bet. He's a tag. We're both full

stacked. His 5 Bet Range is strong JJ+/AK (the top 3%) His 3 Bet range is 9%.Our opponent never calls a 4bet he just folds or 5bets.Lets look at the bet sizings in a blind on blind situation. We steal from the Small Blind 3bbs, He 3Bets from the Big blind to 10bbs, We 4bet to 25bbs with the intention of calling.

EV = (F*P) + (1-F)*(-L*V)+(W*H)

Let's break it down. We have some fold equity here. If he folds to our 4bet this is great. But we also know we will call in this instance too, so we have to calculate what happens when the hands clash.

F= he 3bets 9% and shoves 3 % Fold Equity = 0.67P= If he folds to the 4 bet. There was 3bbs from our pfr he had 3bet to 10bbs total. 13bbs in the pot.

1-F we know this is 1-0.67 = 0.33

-L= is the cost of our 4bet and 5bet call. (90bbs)V= Villains Equity JJ+/AK vs our JJ 63.5% or 0.635W = amount we win when called +113 (100bbs off our opponent + 10bb 3bet + 3bb pfr)H= Hero Equity 0.365 = 36.5%

using formula 5EV = (F*P) + (1-F)*(-L*V)+(W*H)(0.67*13) + (1-F) * (-90*0.635)+(113*0.365)(8.71) + (0.33) * (-57.15)+(41.25)(8.71) + (0.33) * (-15.9)(8.71) + (-5.25) = +EV 3.46

Semi-Bluff Shoving the Flop or Turn. (pg15)

We can use this similar equation when we shove all in post flop over a bet, on the flop or turn. Shoving as a semi-bluff on the flop or turn can be very profitable in some 3 bet pots, as the preflop 3bettor, or versus the preflop 3bettor. I've noticed regs calling down light in 3bet pots though, so this move is best saved for players who will fold some hands post flop in 3bet pots. As you are likely to get called sometimes.Semi-bluff shoving with overcards with flush draws or gutshots, or open ended straight draws can be very profitable as even when called we have good equity.

Example-We call a big blinds 10bb 3 bet on the button with KcQc. (100bbs dp)

Flop - Jc7d2h

Opp C-Bets 12 into 20.5bb pot. We make the call. (44.5pot)We have good equity with overcards and backdoor flush drawOur opponent may check-fold a hand like 88 on the turn.

Turn – 9c

Opp C-Bets 22bbs, we have a read that he can double barrel bluff inthis spot, and will bet some semi bluffs. New pot is 66.5bbs and wehave 78bbs left. We can shove all in with good equity when calledand we might get some folds which is great. On this board we won'tget many folds obviously. Lets say, he calls all top pair top kickerhands and overpairs and all nut flush draws. We will say he bet -folds AK.

We will give our opponent a 3bet range of 10%, but say he only double barrels, AK as abluff and nut flush draws as a semi-bluff, and top pair top kicker and overpairs. All otherhands he looks to get to showdown with.

Combos -AK- I have one King so he has 12 combos of AK.AJ – There is one J on flop, so he has 12 combos of AJAcTc – One Combo of AT of suited clubsJJ – 3 Combos.QQ – 3 CombosKK – 3 CombosAA – 6 Combos.

Total Combinations in opponents Turn Betting Range = 12+12+1+3+3+3+6 = 40 Combos.

(pg.16)Total combos = 40.

Combos he folds = AK = 12 12/40 = 0.3 (30% of range)

He calls with all other combos. -

Now we need to use a software like pokerstove to look at our equity when called vs the other hands.AA = Hero Equity – 26.14%KK = Hero Equity – 27.27%QQ = Hero Equity – 34.09%AJ = Hero Equity – 40.34%

To make our calculation easier lets have KK/AA together and get an average equity and also have QQ and AJ together to find an average equity.

AA/KK – Hero Equity 26.5% 9 Combos from 40 total combos. 9/40 = 0.225 of range. (22.5%)QQ/AJ – Hero Equity 39.09% 15 Combos from 40 total combos. 15/40 = 0.375 of range.(37.5%)JJ – Hero Equity 22.73% 3 Combos from 40 total . 3/40 = 0.075 of range (7.5%)AcTc Hero Equity 20.45% 1 Combo from 40 total. 1/40 = 0.025 of range. (2.5%)

Remember he folds 30% of his range too w/ AK. So if we add up all the percentages they should add up to 100% or add up all the decimal numbers to make 1.

30+22.5+37.5+7.5+2.5 = 100%0.3+0.225+0.375+0.075+0.025= 1

EV = (F*P) + (1-F)*(-L*V)+(W*H)

F= Fold Equity (0.3)P= Money In Pot that we win when they fold (66.5bbs)1-F = 1-0.3=0.7H+V have already been figured out above.L= Maximum Loss (Cost of Shove 78bbs)W= Maximum Win (our opponent covers us, we win 78+66.5 = 144.5bbs

F*P = 0.3* 66.5 =19.95 (We win 19.95 everytime he folds)

1-F = 0.7

Now we have to see what happens when all the hands clash:-

(pg17.)Firstly AA/KK-

0.225(% of AA/KK combos in turn betting range) * (-L*V)+(W*H)0.225 * (-78*0.735)+(144.5*0.265)=0.225 * (-57.33) + (38.2925)0.225 * (-19.0375) = -4.28

Plug these numbers into Formula 4.After plugging in the values for F/P/L/V/W/H look at column I on the spreadsheet and row 20.Insert the decimal number for combos of hands, in this instance 0.225 and in column I row 22the new EV calculation will appear here.

Now we use the same equation for the rest of the hands. All we need to change is the Hero's equity for the AJ/QQ hands. So, in the H column, we input 0.3909, and then change column I row 20 for the combos of hands. So we input 0.375 here. Now we have the new equity for this range of hands.

QQ/AJ = EV +3.36JJ (change H to 0.2273 and change I20 to 0.075) = -2.06AcTc (change H to 0.2045 and change I20 to 0.025) = -0.8

Finally all we have to do is add up these up. Along with the Fold EquityFE = +19.95AA/KK = -4.28QQ/AJ = +3.36JJ = -2.06AcTc = -0.8 = EV of + 16.17

This is about as complicated as it gets for poker math, and if you can understand thisthen you should be good to go and analyse any poker situation.

Bluffing Betting or Bluff Raising the River (pg18)

Now we will look at a spot where we are going to bluff the river and when we get calledwe are sure that we are beat. Eg, we have a 5 high bust flush draw.We are going to bet 14 bbs into a 21bb pot. Firstly we can use formula 6 to find a break even fold %. Input our bet size and pot before our bet. As we can see from the risk/reward chart or from the spreadsheet we need this 2/3rds bet to work 0.4 or 40% of the time for it to be break even. Anything over that and we have a profitable bluff.

Let's analyse this a little more using formula 7.

EV = (F*P) + (1-F) * (-L)

F=Fold EquityP=Money we win when opponent folds. (21bbs)1-F = Opponent Call %-L = Maximum Loss (Cost of our bluff = 14bbs)

EV=(F*21) + (1-F) * (-14)

Now we just need to estimate how often our opponent folds the river by doing some range building and making some assumptions.

eg. On a board of KJ228ss

He calls 12 combos of AK and KQ and calls 9 combos of KJ (21 combos that beat us)And folds 6 combos of QQ to the river bet and 12 combos of AJ (18 combos we fold out)We will say he folds 3 combos of flush draws too. (3 combos of flush draws)

Total combos in range = 42Combos we get called with = 21 21/42 = 0.5Combos we Fold Out (FE) = 21 21/42 = 0.5

EV=(0.5*21)+(1-F)*(-14)(10.5) + (1-F)*(-14)(10.5) + (0.5) *(-14)(10.5) + (-7) = 3.5 EV

Input these numbers into formula 7 to check our workings out.

This formula or method also works for working out profitability of bluff raising the river.

Eg If my opponent raises my 14 bet to 42 . The pot after my bet of 14 into 21 would be 35.Input 35, into formula 6, pot before our bet or raise. And input bet size of 42 in our bet column.We see this raise needs to 0.54 or 54% of the time or more to be profitable.

(pg.19)Using Formula 7.we input 35 into the P column and 42 into cost of bluff column. If we now type in 0.545454 into the F column, we can see that the EV will be very close to zero, as this is our break even fold % in the F column. To find our fold equity in this spot we have to figure out what hands our opponent is betting the river with and see how often we can get him to fold. If we do have a hand with showdown value.

eg, QQ on the KJ228ss board.

If we make the raise with QQ, we have to find enough combos of BETTER hands than QQ that we can fold out. If he calls with worse, this isn't a bluff, but a thin value raise.So if we have a read that our opponent will bet 2 bust flush draws on the river and bet AK/KQ and KJ, but he will only call the river raise with KJ and fold the other hands...

KJ (9 combos)KQ (6 combos, we block some with our QQ and a K blocker on the flop)AK (12 combos)T9s/54s (2 combos)

Total Combos in range = 29 combos.Hands we fold out we beat = 2 combosHands we fold out that are better than our hand = 18 combosHands he calls with that are better = 9 Combos

We ignore the 2 flush draw combos and just use the other combos.Total of combos in range = 27We fold out 18/27 = 0.66 or 66%He calls with 9/27 = 0.33 or 33%

Inputting 0.66 into the F column of formula 7, we can see that if our assumptions are correct the EV of this river bluff would be +8.82.

Calling a River Bet (pg20)

This is fairly simple math up to the previous 2 bluffing calculations. Whenever we arecalling a river bet we will be seeing a showdown. So we just need to use the showdown part of the EV calc used in formula 5. For this calculation we will be using formula 3 and formula 4.

Let's just look at the sizing of the river bet we're facing first and find our break even CALL %.

Opponent bets 1/2pot on the river. 10bbs into a 20bb pot.Input 10 into the Cost of our call column in formula 3.If our opponent bets 10 into 20 then the pot size after his bet is now 30.We input 30 into Pot aft Opp Bet column in formula 3. We see that our Break Even call % for calling our opponents half pot river bet is 0.25 or 25%. You can match that with the risk/reward chat on pg3.Check the calling column.

Now we move onto formula 4.

EV=(-LV+WH) (As you can see this is the last part of the main big equation from pg11).

-L = Maximum Loss or Cost of Call (10bbs)W = Maximum Win or Pot before we Call (30bbs)H = Hero Equity when we call. (input 0.25 to H column to get an EV of exactly zero!)V = Villains Equity when we call.

To figure out our equity, we need to do some range building.Using our previous example lets say our opponent bets KQ and KJ on the KJ226s boardand we have AK.We beat his combos of KQ, but lose to his combos of KJ.KQ combos = 8 (we block 1 K and the flop blocks another K)KJ combos = 6 (2 Kings blocked and 1 J blocked.)

Total combos 14.we beat 8 and lose to 6.

Combos we beat = 8/14 = Hero Equity 0.57Combos we lose to = 6/14 = Villains Equity 0.43.

To figure out these equities we can also use poker stove or a poker equilator. If we haveAdKh and the board texture is - KcJs2s2h8cand we give our opponent a range of any KJ or any KQ, you will see the equity for AK turns out at 57.14% or 0.57.

(pg21)Now we simply add 0.57 to the hero column of formula 4.

EV = (-L*V)+(W*H)= (-10*0.43)+(30*0.57)= (-4.3) + (17.1) = EV +12.8

Thin Value Betting The River.

When thin value betting the river, we need to compare the EV of betting with the EV ofchecking. It's a lot easier to evaluate when we have position and our opponent checks to us.

Scenario-Opp checks to us on the river.

Combos in his range = 10We can beat 7 of those combos.

Pot = $10EV of checking = 7/10 = 0.7 * 10 = $7 EV of checking = $7.

Pot = $10Our opponent will call a $10 bet with 5 combos.We beat 2 combos and lose to 3 combos.

(F*P)+(H*W)+(V*L)

Breaking this down we see -

F=How often he folds when we bet – 5/10 = 0.5P=How much we win when he folds - $10 pot.

H=Hero Equity or how often he calls with worse. 2/10 or 0.2W=Maximum Win when he calls = $20 (pot+call of our bet)

V=Villains Equity or how often he calls with better. 3/10 or 0.3L=Maximum Loss. How much we lose when he calls with better. $10 cost of river bet.

(0.5*10)+(0.2*20)+(0.3*(-10)=(5)+(4)+(-3) = 6 EV of Value Bet = +6

The EV of Checking Back = +7.So in this spot it is more profitable to check back as we win more.

(pg22).Checking – Betting =7-6 = +1 to checking.

Looking at formula 8 we see that in Part 1 of the calculation we need to input these stats into the individual columns.

H= Enter number of combos in villains range we can beat, or Hero's Equity.T = Enter TOTAL number of combos in villains river range that he checks to us.

W = Enter the size of our Value Bet.V = Combos in villains calling range that we lose to, or villains equity.L = Cost of our losing value bet.

In column E Row 46 we will see the Expected Value of Betting over Checking.

To check and see if our math works out we can look at just our opponents calling range.In this instance there are just 5 combos in his calling range. 2 we beat and 3 we lose to. We just need to evaluate this calling range with the cost of our river bet and not take into account the full pot.

Combos of hands we beat 2/5 or 0.4 , combos of hands we lose to 3/5 or 0.6.0.4 * ($10) + 0.6 * (-$10) =(+$4) + (-$6) = -$2

Multiply this -$2 by the % of his calling range. Call range is 5/10 or 0.5

0.5 * -$2 = -$1

This is fairly complicated but I believe using the Spreadsheet makes it much easier.

This can all be simplified. If we bet and can find 1 more combination of hands that we can beat up to the combinations we lose to, then betting is best. If in doubt and you're playing a passive player who doesn't bluff raise, our line should be to just bet.

ExampleVillain has 100 combos of hands in river range.51 combos are worse than our hand and 49 are better. This is a profitable thin value bet.

Generally, the only time we should perhaps need to check the math of our thin value bets is, if we go through a spell of getting called by better hands on the river a lot.

I hope this reference book and spreadsheet helps you to get through poker hand analysisquicker so you get more time to crush the poker tables!

-JCP.(jcp444@yahoo.co.uk)

Acknowledgments-

I've been playing poker for about 5 years. Now Playing 100nl /200nl cash.I have picked up these math skills along the way from various sources.

– Outstanding Poker. When I joined them I was a losing player, but after being an OP member for a couple of years I was able to beat 25nl and I am now also a guest coach there. www.outstandingpoker.com

– Drag the Bar. Another online poker training site, with some high class coaches. The coaches who I learnt some math from were mainly 'Q-Tip' (Owen Gaines)who has a great math book out called 'Poker Math That Matters'.Also 'Vulcans' (Paul Ratchford) He also has a book out called 'Exploitive NLHE'.www.dragthebar.com http://www.thepokercapitalist.comhttp://qtippoker.com/

– I also completed the Iron Lion course on www.pokerzion.com . This course had plenty of solid poker math material.

(author-John Sutcliffe)