This document discusses the use of mathematics in poker. It begins by listing some top earning poker players and their career tournament winnings. It then provides some poker records, such as the player with the most World Series of Poker bracelets. The document then discusses some key poker mathematical concepts like outs, odds, pot odds, implied odds, and expected value. It provides examples and calculations to demonstrate how to apply these concepts in poker situations. Finally, it works through a full example hand to show how to analyze pot odds on a draw by calculating outs and pot odds percentages.

1. Poker in Numbers

2. Introduction
Depending on who you ask, Poker is considered a game of many things:
•
•
•
•
Skill?
Psychology?
Luck?
Mathematics?
A poker deck contains only 52 cards in a finite number of combinations, therefore it is
entirely possible to factor in mathematical equations to assist you in certain situations.
Over these slides, we aim to improve your understanding of mathematics in poker by
helping you understand all of the different ways you can use it to your advantage.

3. All Time Money List…
Before we get to the crunch of the numbers, let’s have a look at some ‘maths’ facts in
poker…
The “All Time Money List measures the live earnings for tournament poker, the
following players all have tournament winnings in excess of $15,000,000
Antonio Esfandiari (U.S.A) :
Sam Tricket (England) :
Daniel Negreanu (Canada) :
Erik Seidel (U.S.A) :
Phil Hellmuth (U.S.A) :
Phil Ivey (U.S.A) :
John Juanda (Indonesia) :
$25,578,279
$19,887,332
$18,539,857
$18,425,444
$17,972,724
$17,649,220
$15,113,768

4. Record Numbers
Most World Series of Poker (WSOP) Bracelets
Phil Hellmuth (13)
Oldest WSOP Participant
Jack Ury (97 Years Old)
Most Expensive Poker Tournament
The Big One for One Drop ($1,000,000 buy in)
Biggest Single Live Cash Pot
Tom Dwan vs Phil Ivey ($1,108,500)
Longest Live Poker Session
Phil Laak (115 Hours, 5 min breaks every hour)

5. Record Numbers
Online poker giants PokerStars recently celebrated a milestone in poker, with
the 1,000,000,000th hand dealt on the site (1 Billion).
The hand awarded over $1,000,000 in prize money to people playing the
same game/stakes and a massive share of the money was awarded to
everyone dealt into the milestone hand:
The hand was won by
a pair of J’s!!
Here is a picture for you:

6. Maths in Poker
Here are all of the mathematical principles we will be covering over the next set of
slides:
•
•
•
•
•
•
•
•
Outs
Odds
Pot Odds
Implied Odds
Reverse Implied Odds
Expected Value
Fold Equity
‘Imagined Money’
– Sklansky Dollars
– Galfond Bucks

7. Outs
An ‘Out’ is the name of a card (or number of cards) that you would need to hit in
order to improve your hand
So for example:
If you held A♦ K♦, and the flop came down: 2♦ 5♦ 7♠
You have a diamond flush draw; meaning you need 1 more diamond to hit your hand:
(We remove the diamonds that are in our hand, or on the flop)
A♦ 2♦ 3♦ 4♦ 5♦ 6♦ 7♦ 8♦ 9♦ T♦ J♦ Q♦ K♦
3♦ 4♦ 6♦ 7♦ 8♦ 9♦ T♦ J♦ Q♦ = 9 ‘Outs’

8. Outs – Some Examples…
1) Hole Cards: J♠ Q♣
-
Flop: A♦ T♣ 2♥
In this hand, we have a straight draw, we need any K in the deck to make our hand:
K♠ K♣ K♦ K♥ - Thus, we have 4 outs!
2) Hole Cards: 8♥ 7♥ - Flop: 6♥ 9♥ A♦
In this hand, we have a straight-flush draw! In this situation, we organise our list of
outs by the ones which will improve our hand the most.
We have outs to make the following hands:
Straight Flush (The Best)
Flush
Straight
T♥ 5♥
A♥ K♥ Q♥ J♥ 4♥ 3♥ 2♥
^Straight Flush
^Flush
T♦ T♣ T♠ 5♦ 5♣ 5♠ = 15 Outs!
^Straight

9. Outs – Examples in Televised Poker
2012 World Series of Poker
Having 18 outs is rare, with just
shy of half the deck containing
cards he needs to win.
Luckily for Merson, he hit a 9 on
the river for a straight!
EPT Grand Final, May 2013
Busquet found himself on the
wrong end of an AA vs KK.
In this situation, he only had 2
outs, the 2 remaining K’s.
Unfortunately, the river bricked
and he was eliminated.

10. Odds
Simply knowing how many outs you have only gives a slight indication of how likely
you are to hit a hand:
i.e. if you have 18 outs, you are more likely to hit a hand than if you have 2 outs…
There is a simple calculation you can do which converts the number of outs you have
into a % value chance of actually hitting your hand; much more useful!
If you are on the flop, and have 2 chances to hit your card:
Multiply your number of outs by 4
If you are on the turn, and have 1 chance to hit your card:
Multiply your number of outs by 2
This calculation isn’t exact and can be off by around 1% for some situations, but the
simpler method allows you to work it out in the moment.

11. Odds - Examples
Taking examples from the previous slide:
1) Hole Cards: J♠ Q♣ - Flop: A♦ T♣ 2♥
In this hand, we have a straight draw, we need any K in the deck to make our hand:
K♠ K♣ K♦ K♥ - Thus, we have 4 outs!
4 outs, we are on the flop. Meaning we multiply our number of outs by 4.
(Outs) x 4 = % Chance
(4) x 4 = 16% Chance
We have a 16% chance of hitting our hand, not a great amount 

12. Odds - Examples
Taking examples from the previous slide:
2) Hole Cards: 8♥ 7♥ - Flop: 6♥ 9♥ A♦
In this hand, we have a straight-flush draw! In this situation, we organise our list of
outs by the ones which will improve our hand the most.
15 outs. But lets imagine we are on the turn…
Turn: 6♥ 9♥ A♦ 2♠
We still have 15 outs but only 1 chance to hit; this means we multiply by 2, not 4.
(Outs) x 2 = % Chance
(15) x 2 = 30% Chance
Even though we have 15 outs, trying to hit a draw on a single card is very difficult, we
hit our hand just shy of 1 in 3 times.

13. Outs
Just Turn
Just River
Turn + River
Example
1
2.13%
2.17%
4.26%
Improving trips to quads
E.g. AA vs KK on AK2 board, the KK hand needs to hit the final K in the deck to take the lead
2
4.26%
4.35%
8.42%
Improve a pair to trips
E.g. JJ vs 55, the 55 hand needs to hit one of the 2 remaining 5’s to take the lead
3
6.38%
6.52%
12.49%
When your hand is dominated
E.g. AK vs AQ, the AQ hand will need to hit one of the 3 remaining Q’s to take the lead.
4
8.51%
8.70%
16.47%
Inside Straight Draw
E.g. 4, 5, x, 7, 8. You need one of the remaining 6’s to make a straight.
5
10.64%
10.87%
20.35%
1 Pair with an Overcard
E.g. KQ on 2QA flop, you need either one of the 3 remaining K’s or 2 remaining Q’s
6
12.77%
13.04%
24.14%
2 Overcards
E.g. AK on a 357 flop, you need either one of the 3 remaining A’s or K’s
7
14.89%
15.22%
27.84%
Improving trips to quads or a full house
E.g. 77 on a 567 flop, you need either a 7 to make quads, or 3 remaining 5’s or 6’s to make a full house.
8
17.02%
17.39%
31.45%
Open Ended Straight Draw
E.g. 67 on a K45 flop, you need either one of the 4 remaining 3’s or 8’s to make a straight
9
19.15%
19.57%
34.97%
Flush Draw
E.g. xx of diamonds on a flop with 2 diamonds, you need one of the 9 remaining diamonds
10
21.28%
21.74%
38.39%
Concealed Pair with a Straight Draw
E.g. 88 on a 679 flop, either of the 2 remaining 8’s will give you quads, along with the 4 remaining 5’s and T’s which will
improve your hand to a straight
11
23.40%
23.91%
41.72%
Overcard with a Straight Draw
E.g. A9 on a 678 flop, either of the 3 remaining A’s will give you a pair, along with the 4 remaining 5’s and T’s which will
improve you to a straight
12
25.53%
26.09%
44.96%
Flush Draw with an Inside Straight Draw
E.g. 89 of spades on a 752 flop with 2 spades. Any of the 9 remaining spades will give you a flush along with the 3
remaining 6’s giving you a straight.
13
27.66%
28.26%
48.10%
Improving a pair/Straight Draw into 2 pair or better
E.g. JQ on a 9TJ board. Any of the 4 remaining K’s or 4 remaining 8’s will give you a straight, any of the 2 remaining J’s
will give you trips and any of 3 remaining Q’s will give you 2 pair
14
29.79%
30.43%
51.16%
Straight Draw with Overcards
E.g. 9T on a 478, with any of the 4 remaining 6’s or 4 remaining J’s giving you a straight and any of the 3 remaining 9’s
or 3 remaining T’s giving you a pair.
15
31.91%
32.61%
54.12%
Flush Draw with Overcards
E.g. AK of diamonds on a 348 board with 2 diamonds. Any of the 9 remaining diamonds will give you a flush, along with
the 3 remaining A’s and the 3 remaining K’s to give you a pair.

14. Pot Odds
Pot Odds are the name for the ratio of a players bet size compared to the total size of
the pot.
They are called ‘Pot Odds’ because they are the comparative odds between:
a) The bet you need to call to be able to win the pot
b) The total size of the pot after your opponents bet is factored in
Pot Odds = Pot Size: Bet Size
Here is an example:
You are playing in a game of No Limit Hold’em, and the pot currently stands at $50.
Your opponent bets $25 – This means the pot now stands at $75
Pot Odds = Pot Size: Bet Size
= $75:$25
= 3:1 (Simplified the Ratio)
You can convert that amount to a % too, in this case 3:1 = 33%.

15. Using Pot Odds on Draws
Using Pot Odds to calculate whether to call on a draw is one of the most vital aspects
of a player who is looking to incorporate more mathematics in their game.
Being able to do this counts on knowing the following in the situation:
1) Your current outs, and the % chance of hitting your hand
2) The current pot odds, again in % format
The 2 values can help you decide whether you are getting the ‘correct’ pot odds to
call on a draw.
If you have pot odds of 33% (1/3) then you would need to win this ‘hand’ 1 out of
every 3 times to ‘break even’ and win your money back. Once you have calculated
the chance of hitting your hand you can compare it to the pot odds.
If the chance of hitting your hand is more than 33%, you will win money in the long
term because you will be winning the hand more often than you need too to ‘break
even’. On the flip-side, if the chance of hitting your hand is less than 33%, then you
will lose money over the long term because you wont be winning it on average 1 in 3
times.

16. Using Pot Odds on Draws
Profitable Call:
Pot Odds – 3:1 (33%)
vs
Drawing odds – 2:1 (50%)
Unprofitable Call:
Pot Odds – 2:1 (50%)
Vs
Drawing Odds – 3:1 (33%)
In this situation,
Our Pot odds tell us, we only need to win this hand 33%
of the time on average to break even.
Our Drawing Odds tell us that we will hit our hand on
average 50% of the time. Thus, this is a profitable call
In this situation,
Our Pot odds tell us, we only need to win this hand 50%
of the time on average to break even.
Our Drawing Odds tell us that we will hit our hand on
average 33% of the time. Thus, this is an unprofitable
call.
Being able to work out the difference and whether a call is profitable or not can help
you make good calls when on draws.

17. Putting it all Together!
Now we are going to run through a big example hand to put everything you have
learned so far into practice. On this slide we will present the problem, on the next
slide we will work out the solution!
You are playing a No Limit Texas Hold ‘em Tournament.
Blinds are t100/t200, no ante. Your stack: t12,000 chips.
Action is folded around to a player in middle position, who raises to t500. It is then
folded around to you on the button, you look down at your hand: 7♠ 6♠.
You elect to just smooth call the bet, and both blinds fold. The pot now stands at:
t1300 chips.
Flop: 8♦ 9♠ A♥
Your opponent bets t800 chips, to make the total pot t2100. Is this a profitable call?

18. Putting it all Together! Pt. 2
First things first, lets work out the ‘outs’:
Hand: 7♠ 6♠
Flop: 8♦ 9♠ A♥, We are on a straight draw
This means we have 8 outs: 5♠ 5♣ 5♦ 5♥ T♠ T♣ T♦ T♥
Since we have 8 outs, and 2 chances to hit our cards, we can work out our % chance
Turn+River = Outs x 4
=8x4
= 32% (The calculated amount is 31.45%, but this is close enough)
Now we have our drawing odds we can now calculate our pot odds
Pot Odds = Pot Size : Bet Size
= 2100 : 800
= 2.63 : 1 (Simplified)
= 38% (To Calculate %: 1 ÷ 2.63)

19. Results…
We now have our 2 requirements:
Drawing Odds: 32%
Pot Odds: 38%
Our drawing odds inform us that we will hit our hand 32% of the time, but for this call
to be break even we need to win this hand 38% of the time or more.
Unfortunately, making this call would be unprofitable.. 
But you should be able to work out pot odds by now! 

20. Implied Odds
When discussing Pot Odds, we showed you an example hand that involved a straight
draw on the flop, but after a quick calculation we deduced that the call was in-fact
unprofitable.
However…
Sometimes, a draw might be unprofitable when looking at immediate pot odds, but
certain situations might make a call profitable when you take into account implied
odds.
Implied odds adjust pot odds based on how concealed your draw is, and the
likelihood you can get paid off by an opponent if you hit your draw.
This factors in a few things:
1) Opponents Table Image
2) Board Texture
3) Your immediate Pot Odds

21. Implied Odds – Cont.
Pot Odds only take into account the immediate bet and the odds of hitting your hand,
they don’t take into account future betting rounds.
The ability to earn exponentially more chips on future betting streets if you hit a draw
balance out what might be immediately unprofitable pot odds.
You have good implied odds if you are more likely to make money on future betting
rounds, and oppositely you will have bad implied odds if you are unlikely to make
money on future betting rounds.
The ability to discern between situations where you have good implied odds and bad
implied odds is vital to deciding whether continuing on a draw is necessary and has
potential.
Regardless of your Implied Odds, If you are getting very bad pot odds, then you
should fold regardless. E.g. if your drawing odds are 10% but your pot odds are 50%
then you should just dump your hand.

22. Situational Implied Odds
Opponents Table Image
If you are playing against an aggressive opponent who bets frequently, you can be
comfortable that they won’t ‘shut down’ and stop betting on future streets which
allows you to earn , have good implied odds.
If you are playing against a passive opponent who doesn’t bet often, you wont be
able to extract enough value on future betting streets, and thus you have bad implied
odds.
Draw/Board Texture
If you are on a disguised draw (like an inside straight draw) that wont be obvious
when it comes in, your opponent likely wont think you have made a hand and thus
they will keep betting and you have good implied odds
However, if your draw is obvious, (for example, 4 cards to a flush/straight) then the
board texture is scary to your opponent who will know you are drawing and will likely
stop betting, thus you have bad implied odds

23. Implied Odds - Example
Taking our Pot Odds Example:
Hand: 7♠ 6♠
Flop: 8♦ 9♠ A♥, We are on a straight draw
This means we have 8 outs: 5♠ 5♣ 5♦ 5♥ T♠ T♣ T♦ T♥
Opponent bets t800 into a pot of t1300, making the pot t2100.
We worked out our Pot Odds were 38% and our drawing odds were 32%
We can already assume that our opponent is aggressive as he is making a sizeable
bet, and he also raised pre-flop.
Our draw is semi-disguised. If the turn changes the board texture to: 8♦ 9♠ A♥ 5♦ our
hand is disguised. If the board texture changes to: 8♦ 9♠ A♥ T♦ this is a bit ‘scarier’ to
our opponent as this hand helps both QJ and 67.
Regardless, our Pot Odds and Draw Odds are fairly close, our opponent is aggressive
and our draw is disguised. This means we have profitable implied odds.

24. Implied Odds - Table
Implied Odds
Profitable
Un-Profitable
Pot Odds
Good
Bad
If you have good implied odds and If you have good pot odds but your
the correct pot odds!
draw is obvious, you can still call as
your immediate odds are profitable.
If you have good implied odds but If you don’t have profitable Pot Odds
unprofitable pot odds, you can
or decent implied odds then you
call because you can make enough should fold your hand.
money on future betting streets
to balance the unprofitable call.

25. Reverse Implied Odds
Taking a break from the very mathematical equation we just calculated, lets look at
Reverse Implied Odds.
Just a recap, Implied Odds are the odds of you earning exponentially more chips on
future betting rounds. Based on the visibility of the draw and our opponents table
image.
Reverse Implied Odds come into effect if the draw we are on might not be the best
hand if the card(s) we need does actually come.
Simple Example
If we are holding 5♠ 6♣ and the flop is 7♠ 8♥ T♥. Any 9 or 4 makes our straight. However, if a 9
comes in, someone could easily have a J for a higher straight. If it is the 9♥ then an opponent could
have made a flush that beats our straight.
In general, Reverse Implied Odds are when you are not drawing to the nuts (The best hand
possible).

26. Reverse Implied Odds - Example
Your Hand: 4♥ 6♥
Flop: 7♠ 8♠ 9♥
We are on a straight draw: Imagine our opponent bets t600 into a pot of t1000 chips.
We need to call t1600 to chase our draw. A quick calculation shows us that we have 8 immediate
outs and 2 chances to hit them: 32% chance.
We have 600/1000 pot odds, simplified: 60%. Meaning we are not getting the correct pot odds.
Our implied odds are reversed in this situation. A 5 or a T give us a straight, but someone could
easily have a higher straight. If a spade comes too, then our opponent could hit a flush, beating our
straight.
Here are our 8 outs: 5♥ 5♦ 5♣ 5♠ T♥ T♦ T♣ T♠
Our 2 spade outs might give our opponent a flush, so we can call them “Dirty Outs” and cross them
off our list. Also, considering we could already be losing to TJ and if a 5 or T hits we could still lose
to a J or a Q. This draw has very Bad Reverse Implied Odds and thus, should be folded.

27. Expected Value
Expected Value is a term commonly associated with probability and odds… this is
perfect for poker!
Expected Value is the amount you expect to win or lose depending on a situation and
the odds around it.
+EV = profitable/good choice. This is one that will win you money over the long term.
-EV = unprofitable/bad choice. This is one that will lose you money over the long term.
Simple Example
You and a friend make a wager: You roll a dice; if the number is even then you win $10
from your friend, if the number is odd you lose $10.
Each die has 3 even numbers and 3 odd numbers. This means that you will win 50% of
the time and lose 50% of the time.
50% of the time you are winning $10 and 50% of the time you are losing $10. Over an
infinite number of hands you will be breaking even. This is neutral EV (~EV)

28. Calculating Simple EV
EV Example Calculation
Your friend is feeling lucky. He decides to up the wager: Every time the dice has an
even number you win $20, every time the dice has an odd number you lose $10.
Our odds are still 50/50 in this situation, but now 50% of the time we win $20, and
50% of the time we lose $10.
Expected Value = (Odds of winning x winning wager) – (Odds of losing x losing wager)
= ( 50% * $20 ) – ( 50% * $10 )
= ( $10 ) – ( $5 )
= +$5
In this situation, every time we roll the die, we are expecting to win $5.
This won’t happen, because we can only win $20 or lose $10.
But on average we expect to win $5 per roll. So roll that die as many times as possible!

29. Calculating EV
EV Example – From the Felt
This example is taken from a cash game: You are playing $1/$2 No Limit at your local
casino. You are on the flop against 1 very short stacked opponent, and the pot stands
at $13.
Hole Cards: A♣ Q♥
-
Flop: A♠ 5♥ 6♥
Your opponent bets all in for his last $12 and picks up his cards; he then accidentally
drops them face up, revealing 7♥ 8♥ for a straight flush draw.
Now, what do we do?
We technically have the best hand, so this isn’t a situation where we are needing to
consult Pot Odds. However, we can calculate whether this call would be +EV or –EV
over the long run.
We use a tool called ‘PokerStove’ for this

30. PokerStove
PokerStove is an equity calculation tool which we can use to help us make more
complicated mathematical decisions.
It works out the chance of winning depending on 2 different hands or ranges of hands.
Equity Calculator:

31. Continuing with our EV Calculation
Hole Cards: A♣ Q♥
-
Flop: A♠ 5♥ 6♥
Your opponent bets his last $12 and picks up his cards; he then accidentally drops them
face up, revealing 7♥ 8♥ for a straight flush draw. The pot now stands at $25.
PokerStove:
In this situation, we have 48% equity (chance to win), we can now work out the EV from
this decision:
Winning Wager: +$37 (Total size of pot if we call)
Losing Wager: -$12
Expected Value = (Odds of winning x winning wager) – (Odds of losing x losing wager)
= ( 48% * 37 ) – ( 52% * 12 )
= (17.75) - (6.24)
= +11.51
Although we either win $37 or lose $12. We make $+11.52 on average per situation.

32. Variance in Poker
The best way to be a winning poker player is to always make decisions that are +EV,
i.e. moves that put you in situations where you expect to make a profit on average.
However, as we (hopefully) demonstrated in the previous slide, the outcome of a
situation is either: You win or You lose.
These immediate outcomes are decided by short-term luck. There is always a chance
that you can lose when you are massive favourite to win; that’s just the nature of a
randomly shuffled deck of cards.
On average you will win (x) amount from a situation over a long enough sample of
hands, but each individual hand will have either a win or a loss.
Therefore you shouldn’t base your attention on whether you won or lost a single
hand. Calculate whether you made a +EV call or not; if you did make a +EV call but still
lost you shouldn’t worry as the variance line will balance out. Focus on playing +EV
poker and the profits will follow.

33. Variance in Poker
$300.00
$250.00
$200.00
$150.00
Average
$100.00
Per Hand
$50.00
$0.00
0
2
4
6
8 10 12 14 16 18 20
The red line indicates a win/lose on a hand per hand basis, whereas the blue line
indicates the consistent profit ($11.51) we expect to win based on our EV
calculation.
As you can see, the 2 lines follow a similar pattern. This shows that even though you
might win or lose a few hands in a row, after a long enough sample, the 2 lines will
follow the same pattern.

34. Fold Equity
Fold Equity is another principle we will be exploring.
It revolves around the odds and probabilities that opponents will fold to a shove, and
how much money we can earn from shoves by getting folds from our opponents.
There is a formula to work out how much profit you could expect to make from
shoving a hand. This takes into account the times your opponents fold, and the times
they call and you either win or lose. It is very similar to the EV calculation, but with 1
crucial addition
To calculate all this, we are going to use our friend Pokerstove again!
But, for reference, here is the formula.
Profit = (% time opponent folds * equity in pot when fold * profit made)
+ (% time call * equity in pot * profit made)
+ (% time call *reverse equity in pot * money lost)

35. Fold Equity in Tournaments
In tournament poker, Fold Equity is especially important considering the rising blinds.
You will find yourself in situations where your stack gets short and you have to start
shoving all in with hands that you don’t want to be called with.
At this point, you need to keep a very close eye on your stack size, as your fold equity
will decrease as your stack gets dangerously short.
At this point, the amount you will be expecting to earn from an opponent is they fold
is decreasing, as they will be calling you with a much wider range and thus, more
often.
On the next screen we have a graph that illustrates how you fold equity decreases as
you stack shrinks. The x-axis scale is set in BB’s.
Just a note: You should also be considering your fold equity if you are shoving a value
hand and want to be called. I.e. Don’t shove AA with 50 big blinds as you wont get
called!!

36. Fold Equity Chart - Tournament
When your stack is:
20+ bb =
Opponents would only ever call a shove with the
top 5-10% of their range, thus, you have a lot of
fold equity. But you shouldn’t be shoving this many
chips anyway as you have enough to play proper
poker.
20-10 bbs =
Opponents start to call shoves with a wider range
as you start to do less damage to their stack
should they call and lose.
10-5 bbs =
Your fold equity starts to drop drastically as your
shoves have less and less weight behind them.
Opponents will call with a very wide range .
0-5bbs =
You have no fold equity at this point, opponents
are only folding absolute junk to a shove
(27, 83, 82 etc.).
Fold Equity
80
60
40
Fold Equity
20
0
20 17 14 11 8 5 2

37. Fold Equity Example
We are playing an online MTT. Blinds are t1000/t2000 with a t200 ante.
Your Stack: t16000, Average stack = t86000.
Action is folded around to you on the button, there is only a big blind in this hand due to a recent
elimination. The player in the big blind is on an average stack.
You look down at K♠ 5♠
Lets look at how successful a shove here would be:
Profit = (% time fold * equity in pot when fold * profit made) + (% time call * equity in pot * profit
made) + (% time call *reverse equity in pot * money lost)
Lets break this down into the 3 sections and tackle each at a time:
(% time fold * equity in pot when fold * profit made)
We only know how often our opponent folds when considering hand ranges. If our opponent was
fairly tight we can assume they are only calling our shove with: AA-66, AT+, A8s+, KQ. This
accounts to 10.7% of the total hand combinations. Thus, they are folding 89.3% of the time.
Our equity in the pot becomes 100% of they fold (we’re the only one left), and the profit we make
is the big blind + antes.
( % time fold * equity in pot when fold * profit made)
( 89.3% * 100% * 3800 )
( 0.893 * 1 * 3800 )

38. Fold Equity Example Cont.
Lets look at the next section:
(% time call * equity in pot * profit made)
Since we worked out our opponent will be calling with 10.7% of hands, that’s the first point done.
Now, with our individual hand against our opponents range, we can work out our equity using
PokerStove:
We have 33.17% equity against that range of hands.
We know that if we are called and win, we double our stack (t16000*2) + the blinds/antes (3800)
(% time call * equity in pot * profit made)
( 10.7% * 33.17% * 35800 )
( 0.107 * 0.3317 * 35800 )
The final Section:
(% time call *reverse equity in pot * money lost)
Reverse Equity is our opponents equity in the pot (66.826%) and we lose our whole stack if we are
called and lose (t16000)
(% time call * reverse equity in pot * money lost)
( 10.7% * 66.83% * -16000)
( 0.107 * 0.6683 * -16000)

39. Fold Equity – Putting it All Together!
So, we have our 3 components!
Profit = (% time fold * equity in pot when fold * profit made) + (% time call * equity in
pot * profit made) + (% time call *reverse equity in pot * money lost)
Profit = ( 0.893 * 1 * 3800 ) + ( 0.107 * 0.3317 * 35800 ) + ( 0.107 * 0.6683 * -16000)
Profit = (3393) + (1270) + (-1144)
Profit = +3519
So, shoving here would be profitable!! (Phew!)
As you can see from the individual numbers, we make the most of our profit from our
opponent folding. Just a note, keep an eye on your fold equity and try your best to
make profitable shoves.

40. Imagined Money
Many of mathematical principles we’ve discussed can be applied mid game to help
you make decisions to either call/fold or shove all in.
Now, we are going to take a look at some theories you can use when you are going
through post-game analysis.
These theories are based around the principle of ‘Imagined Money’ or ‘Fake Money’.
The ‘Imagined Money’ principles show you whether certain actions you take will be
profitable over an infinite number of hands and they don’t take the immediate
outcome into effect.
Thus, analysing your game with ‘Imagined Money’ is perhaps one of the best ways to
show if you are playing profitable poker or not.

41. David Sklansky
David Sklansky is a professional poker player and is sometimes referred to as the
founder of mathematical poker.
He has written numerous poker books about maths and some are still heralded as
some of the best that have ever been written.
Poker Resume:
Bracelets: 3
Money Finishes: 23
Highest ME Finish: 27th, 1988
His 14 poker books can still be
found in many retailers and online.

42. What are Sklansky Dollars?
Sklansky determined that you can calculate whether a call is profitable or not by
comparing the following factors:
1)
2)
3)
Your equity in the pot when you make a call/shove
Your ‘share’ of the pot compared to your equity
Comparison of your initial investment and your equity in the pot
This method is relatively simple and actually makes a lot of sense when thinking in
literal terms.
If you make a call where you have 80% equity in a pot of $100, over enough hands you
would be earning $80 on average (Remember Variance! We discussed this earlier)
If this compared to your original investment brings a positive number, then you can be
sure it is profitable and thus, very good poker!! 

43. Sklansky Dollars - Calculation
Here is how you calculate Sklansky Dollars
1)
Using Pokerstove or a similar tool plug in your hand, your opponents hand and if
necessary the community cards available.
2)
Work out the total pot size at the end of the hand.
3)
Take your equity calculation (from Pokerstove) and work our your share of the
total pot size. (If you have 60% equity in the hand, what is 60% of the total pot
size?)
4)
Compare that amount to the amount you invested in the pot by the action you
are analysing. (I.e., if you want to determine if a call was profitable, work out how
much you invested by making that call)
5)
If the amount you invested is less than your share of the pot, then the positive
amount is the amount of Sklansky Dollars you have made in profit
6)
If the amount you invested is more than your share of the pot, then the negative
amount is the amount of Sklansky Dollars you have lost.

44. Sklansky Dollars – Profit Example
Scenario: We are on the flop, and our opponent has just bet $40 all-in into a pot of
$80, making the total pot size $120. You call with an over-pair to the board.
Our Hand:
T♥ T♣
Opponents Hand:
A♦ 8♦
Board:
8♣ 6♥ 2♥
Pot Size:
$120
Using PokerStove:
We can calculate that we have a 81.41% share of the pot:
(0.8414 * 120) = $100.97
We invested $40 into the total pot, but stand to take a $100.97 share:
$100.97 - $40 = $60.97 profit!
We stand to make $60.97 profit over x amount of hands. This is profitable play, and we
are rewarded with almost 61 Sklansky Dollars!

45. Sklansky Dollars – Loss Example
Scenario: We are on the flop and our opponent has just bet $50 into a pot of
$40, making the total pot size $90. You call his bet:
Our Hand:
6♥ 7♥
Opponents Hand:
K♦ K♣
Board:
K♠ 5♥ 8♥
Pot Size:
$100
Using PokerStove:
We can calculate that we have a 42.121% share of the pot:
(0.42121 * 90) = $37.91
We invested $50 into the total pot, but stand to take $37.91 share:
$37.91 - $50 = -$12.09
Just calling this bet stands to lose us just over $12 every time on average, this this is
unprofitable play and we would lose $12 Sklansky dollars!

46. Phil Galfond
Phil Galfond is an American Poker player, and one of the most successful online
players of all time.
He is considered a pioneer of mathematical poker, adapting the famous Sklansky
Dollars theory into his own method: G-Bucks.
He plays online poker under the screen-name: OMG Clay Aiken and is frequently
found playing the $200/$400 limits online. His winnings on Full Tilt Poker equate to
over $10,000,000; making him one of the highest earners on the site.
Poker Resume:
Bracelet: 1
Final Tables: 9
Cashes: 9
Highest ME Finish: 141st

47. What are G-Bucks?
While relevant, the Sklansky Dollars concept doesn’t take into account times when you
don’t know your opponents hand.
This developed theory by Phil Galfond is very similar to Sklansky Dollars but takes into
account an opponents perceived hand range and not their actual hand.
This allows you to either:
1) Calculate G-Bucks Mid Game
2) Analyse hands where you don’t see your opponents hand (If they fold/you fold)
The theory still uses PokerStove but you need to have some knowledge about hand
ranges to accurately calculate hands.

48. How to Calculate G-Bucks
While similar to Sklansky Dollars, there is a slightly different method for calculation:
1) Analyse the hand history and your opponent to narrow down their playing style.
2) Using their playing style and the events of the current hand, narrow your
opponents range of hands to a reasonable selection of likely choices.
3) You can calculate a value range of hands which your opponent could likely have.
Include semi-bluffs as a part of their range, also try to include a few stone bluffs if
your opponent is likely to be able to pull of such a move.
4) You need to factor in how your opponent plays and how the hand develops, for
example if the board texture is dry, your opponent is tight and he is raising or
shoving you can be certain they have a very narrow range with monster hands like
sets or aces.
5) After you have calculated their range, plug that into pokerstove as your opponents
hand, and compare it to your own in terms of equity, then follow the Sklansky
calculation as normal.

49. G-Bucks Calculation
We are playing online heads up. We have a lot of history against our opponent which
allows us to accurately determine his hand range based on what happens.
We are on the flop against our opponent, the pot is $20 after pre-flop action. On the
flop, we c-bet for $10 and our opponent raises to $30.
Our Hand:
Q♥ J♥
Opponents Hand:
?
Board:
Q♦ 9♦ 8♥
Pot Size:
$60
We can assume our opponent is raising on a semi-bluff due to the nature of the board
texture. Knowing this, we can plug in combinations of 2-pairs/sets and semi-bluff
hands. Our opponent would likely just smooth call most pairs in this spot so we can
rule most of them out.
Here is how we can narrow our opponents hand down: All sets and semi bluffs for
flushes and straights.

50. How to Calculate G-Bucks
Using that knowledge, we can now calculate whether we make G-Bucks from this call:
We have 53.019% equity in the pot
The total pot size: $60, and we need to call $20 more.
Our share of the pot: (0.53019 * 60) = $31.81
We need to call $20 more to win $31.81, this means we are making $11.81 profit over
x amount of hands!
We can also say we have made $11.81 profit in G Bucks!

51. You are now ready to put your skills to the test!