Poker in Numbers

IntroductionDepending 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.

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

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)

Record NumbersOnline 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:

Maths in PokerHere 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

OutsAn ‘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’

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)FlushStraight

T♥ 5♥ A♥ K♥ Q♥ J♥ 4♥ 3♥ 2♥ T♦ T♣ T 5♠ ♦ 5♣ 5 = ♠ 15 Outs! ^Straight Flush ^Flush ^Straight

Outs – Examples in Televised Poker2012 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.

OddsSimply 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!

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.

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

Odds - ExamplesTaking 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

Odds - ExamplesTaking 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.

Outs Just Turn Just River Turn + River Example

1 2.13% 2.17% 4.26% Improving trips to quadsE.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 tripsE.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 DrawE.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 OvercardE.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 houseE.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 DrawE.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 DrawE.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 DrawE.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 DrawE.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 DrawE.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 betterE.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 OvercardsE.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 OvercardsE.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.

Pot OddsPot 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 potb) The total size of the pot after your opponents bet is factored in

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%.

Pot Odds = Pot Size: Bet Size

Using Pot Odds on DrawsUsing 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 hand2) 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.

Using Pot Odds on DrawsProfitable Call:

Pot Odds – 3:1 (33%) vsDrawing odds – 2:1 (50%)

Unprofitable Call:

Pot Odds – 2:1 (50%)Vs Drawing Odds – 3:1 (33%)

Being able to work out the difference and whether a call is profitable or not can help you make good calls when on draws.

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.

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?

Putting it all Together! Pt. 2First things first, lets work out the ‘outs’:

Hand: 7 6 Flop: 8♠ ♠ ♦ 9 A♠ ♥, We are on a straight drawThis 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

= 8 x 4 = 32% (The calculated amount is 31.45%, but this is close enough)

Now we have our drawing odds we can now calculate our pot oddsPot Odds = Pot Size : Bet Size

= 2100 : 800 = 2.63 : 1 (Simplified) = 38% (To Calculate %: 1 ÷ 2.63)

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!

Implied OddsWhen 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 Image2) Board Texture3) Your immediate Pot Odds

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.

Situational Implied OddsOpponents 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 oddsHowever, 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

Implied Odds - ExampleTaking our Pot Odds Example:

Hand: 7 6 Flop: 8♠ ♠ ♦ 9 A♠ ♥, We are on a straight drawThis 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.

Implied Odds - Table

Implied Odds

Pot Odds

Good Bad

Profitable

If you have good implied odds and the correct pot odds!

If you have good pot odds but your draw is obvious, you can still call as your immediate odds are profitable.

Un-Profitable

If you have good implied odds but unprofitable pot odds, you can call because you can make enough money on future betting streets to balance the unprofitable call.

If you don’t have profitable Pot Odds or decent implied odds then you should fold your hand.

Reverse Implied OddsTaking 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).

Reverse Implied Odds - ExampleYour 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.

Expected ValueExpected 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)

Calculating Simple EVEV 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!

Calculating EVEV 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

PokerStovePokerStove 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:

Continuing with our EV CalculationHole 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.

Variance in PokerThe 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.

Variance in Poker

0 2 4 6 8 10 12 14 16 18 20$0.00

$50.00

$100.00

$150.00

$200.00

$250.00

$300.00

AveragePer Hand

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.

Fold EquityFold 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)

Fold Equity in TournamentsIn 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!!

Fold Equity Chart - Tournament

20 16 12 8 4 00

1020304050607080

Fold Equity

Fold Equity

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 ExampleWe 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 )

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)

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.

Imagined MoneyMany 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.

David SklanskyDavid 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: 3Money Finishes: 23Highest ME Finish: 27th, 1988

His 14 poker books can still befound in many retailers and online.

What are Sklansky Dollars?Sklansky determined that you can calculate whether a call is profitable or not by comparing the following factors:

1) Your equity in the pot when you make a call/shove2) Your ‘share’ of the pot compared to your equity3) 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!!

Sklansky Dollars - CalculationHere 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.

Sklansky Dollars – Profit ExampleScenario: 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: Opponents Hand: Board: Pot Size: T ♥ T♣ A ♦ 8♦ 8♣ 6♥ 2♥

$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!

Sklansky Dollars – Loss ExampleScenario: 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: Opponents Hand: Board: Pot Size: 6♥ 7♥ K ♦ K♣ K 5♠ ♥ 8♥

$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!

Phil GalfondPhil 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: 1Final Tables: 9Cashes: 9Highest ME Finish: 141st

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 Game2) 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.

How to Calculate G-BucksWhile 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.

G-Bucks CalculationWe 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: Opponents Hand: Board: Pot Size: Q J♥ ♥ ? Q ♦ 9 ♦ 8♥

$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.

How to Calculate G-BucksUsing 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!

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