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spoonitnow

05-21-2010, 03:50 AM Post subject: spoonitnow's Mathematics of EV Thread

#1 (permalink)

Straight Flush

Join Date: Sep 2005

Location: IRC Chat Room

Posts: 5,406

I'm going to make this thread because I'm tired of answering everyone and their mother's questions about setting up EV equations for cash games. Once this is finished, I'll just start linking people to this thread when they ask me these types of questions. What will follow is a series of math lessons that build on each other to eventually come around and teach you how to do EV calculations that are as complicated as you'd like. As a pre-requisite to this thread, you'll have to know how to multiply, divide, subtract and add real numbers, and also understand how to use decimal numbers (ie: knowing that 0.01 < 0.1, and so on) and exponents (that 3^4 means 3*3*3*3). Hopefully that's it.

This isn't going to be perfect, but I'm trying to lay enough of a foundation that someone can study this and either know what they need to know to do complicated EV calculations, or to at least know what they need to figure out to be able to do complicated EV calculations. Good luck.

Pre-Algebra Section

Introduction

These are a few topics that are usually first taught in a Pre-Algebra class, and they're all related, so I'm grouping them together. Most people will already know some or all of this, or maybe know some of it but aren't sure how each topic relates to the other.

Part 1: Fractions

A fraction is one number divided by another number. For example, 1/3 is a fraction that means one divided by three. It also means some part of a whole. For example, if there were 8 slices of pizza, and you ate 3 of them, then the fraction of the pizza that you ate is 3/8, or three parts out of the whole which was eight parts.

Some fractions can mean the same thing even if different numbers are involved. For example, both 1/4 and 2/8 mean the same thing. If the top and bottom of a fraction can be divided by the same number, then we can "simplify" the fraction by doing the division of the top and bottom by that number. For example, if you have the fraction 6/9, both the top and bottom can be divided by 3. So if you divide the top and bottom by 3, you get 2/3 which is the "simplified" version of 6/9. You can use a calculator to see that 6/9 (six divided by nine) and 2/3 (two divided by three) are the same thing. It's often useful to simplify fractions since you end up with smaller numbers because smaller numbers are easier to deal with.

Poker Example 1: What fraction of a standard 52 deck of cards are spades? There are 13 spades in the deck, and 52 total cards, so the fraction is 13/52. The top and bottom can both be divided by 13, so if we divide the top by 13 we get 1, and if we divide the bottom by 13 we get 4, so the simplified fraction is 1/4. This makes sense because there are 4 suits, and spades are 1 of them.

Poker Example 2: Suppose we have a diamond flush draw on the flop. What fraction of cards can make our hand on the turn? There are 2 cards in our hand and 3 cards on the flop, so that leaves 47 cards that we haven't seen yet. Of those cards, there are 9 diamonds, so the fraction is 9/47, or nine out of forty-seven.

Part 2: Percentages

A percentage is a fraction of 100. If someone says something happens 64% of the time, that means the same thing as saying it happens 64 out of 100 times, or 64/100. Often we'll want to take a fraction like 9/47 from Poker Example 2 above and convert that to a percentage. If you do the division 9 divided by 47 on a calculator, you'll get something like 0.191489. To find the percentage, move the decimal place to the right 2 spaces and you'll get 19.1489%.

One more time. Suppose we want to convert 5/8 to a percentage. First we do the division 5 divided by 8 and get 0.625. Now we move the decimal place two spaces to the right to get the percentage that 5/8 equals: 62.5%.

Poker Example 3: Suppose you have an open-ended straight draw on the turn. What is the fraction of the time you'll make your draw on the river? What percentage is this?

There are 2 cards in your hand and 4 community cards on the board, so that means there are 46 cards left that could come on the river. Since you have an open-ended straight draw, that means 8 of those 46 cards can make your draw, so the fraction is 8/46. To find what percentage this is, we divide 8 by 46 to get 0.173913 and move the decimal place to the right two spaces to get 17.3913%.

Part 3: Ratios

Just like fractions and percentages, a ratio is another way we can use to express how often something happens. What's special about a ratio is that it doesn't express things as part of a whole. Instead, it expresses how big each individual part of the whole is. For example, look at this diagram:

There are 5 balls. As fractions, we could say that 2/5 of the balls are red, and that 3/5 of the balls are blue. We could turn those into percentages to say that 40% of the balls are red, and that 60% of the balls are blue. However, with a ratio, we would say that the ratio of red balls to blue balls is 2:3. Or we could say that the ratio of blue balls to red balls is 3:2.

Ratios can be simplified in the same way as fractions can be simplified. If we have the ratio 20:4, both sides of the ratio can be divided by 4. If we do the division, we get 5:1, which means the same proportion as 20:4. Sometimes we'll simplify ratios to make them have the form X:1, even if that makes X an uneven number. We do this to make it easier to compare different ratios, like comparing pot odds to the odds against hitting a draw.

Ratios are important in poker because it's often easier to think about things like pot odds in terms of ratios instead of fractions or percentages. When things are easier to think about and understand, we have a higher chance of doing something correctly.

Poker Example 4: In Poker Example 2 we showed that the chance of hitting a flush draw on the turn was 9/47 because there are 9 cards out of 47 that make the flush. With a flush draw on the flop, what is the ratio of cards that don't make our flush to cards that do make our flush?

Since there are 47 total cards, and 9 of them make our flush, that leaves 38 cards that do not make our flush. Then the ratio of cards that don't make our flush to cards that do make our flush is 38:9. If we divide both sides by 9, we get 4.22:1. We would say that 4.22:1 (4.22 to 1) are our odds against making our flush.

Poker Example 5: Imagine that you're heads-up on the river in position (he has you covered). After your opponent makes a bet of $50, the pot is $150 and it's your turn to act. What is the ratio of the pot size to the amount you'd be calling if you call?

The pot size is $150, and you'd be calling $50, so the ratio of the pot size to the amount you'd be calling is 150:50, or 3:1. This ratio is important in poker, and is known as your "pot odds".

Algebra Section

Introduction

Algebra is a topic that scares a lot of people, mostly because they haven't taken the time to learn it. I'm going to try to make this as painless as possible. I'm also not going to get into a ton of detail with examples and such. If you want extreme detail without the classroom setting, I'd suggest getting Algebra for Dummies or some other similar text that breaks it down better than most traditional textbooks. Alright, deep breath, here we go.

Part 5: The Distributive Property

This is something that comes up a lot and some people don't fully understand, so I'm making a special place for it here. If you have something that says 6(3+4) normally you'd add 3+4 to get 7 and have 6(7) which is 42.

Side note: If you don't know, 6(7) just means 6 times 7, and similarly 6(3+4) means six times the sum of 3 and 4.

But back when we have 6(3+4), there's another way we can work it out using a relationship between numbers that we call "the distributive property". This says that we can start figuring out 6(3+4) by doing 6*3=18 and 6*4=24 to get 18+24, and then do the addition 18+24 = 42. The name of the property comes from thinking of it as the 6 being "distributed" to the 3 and 4.

One more example. Say we have 4(2-7). Normally we'd do 2-7 to get -5, and 4(-5) = -20. With the distributive property, first we would "distribute" the 4 to get 8-28, and then do the subtraction 8-28 = -20.

Make sure you understand this before you proceed. If you don't understand it, check out the google search for the distributive property.

Part 6: Equations

An equation just means you're saying two things are equal. In real life, if we say two apples and an orange cost $2.50, that's another equation since we're saying that the cost of two apples and one orange at that time in that place "equals" $2.50. In math land, if we say 6 + 4 = 10, that's an equation. In math land, equations need an equals sign. The equals sign divides the equation into two parts, which we usually just call "the left side" and "the right side".

What's really cool about equations are that if (in terms of math) you do the same thing to both sides, it stays equal. So say we have our equation 6 + 4 = 10. If we add 3 to both sides, we get 6 + 4 + 3 = 10 + 3, and since both sides equal 13 now, the equation is still correct.

If we want to multiply both sides by a number, we have to use the distributive property. Say we have 6+4 = 10 and we want to multiply both sides by 3. On the left side we should have 3(6+4) and on the right side we should have 3(10). We have to do this because we want to multiply the WHOLE left and right sides by 3, so we put the whole thing in parentheses. This would leave us with 3(6+4) = 3(10), and both sides equal 30 so we know the equation is still right.

Part 7: Variables

Sometimes we need to do some math on a number but we don't know what that number is yet, so we'll use a letter in place of that number until we figure out what it is. For example, if we want to talk about some generic poker scenario, we could call the pot size by the variable P since we don't know specifically what the pot size is.

Then we can use that variable to talk about bet-sizes and other things. For example, a half-pot sized bet would be P/2. For another example, 30 dollars more than the pot would be P+30, or 20 dollars less than the pot would be P-20.

Twice the pot could be represented as 2*P or just 2P for short. When using multiplication with variables, we usually just write them down beside of each other instead of using a times symbol and it's understood that this means multiplication. So 2P means 2 times P, and something like ab would mean the variable a times the variable b.

Part 8: Using Equations to Solve Unknown Variables

A side note to get you started on the right foot: Say you have 500 chips in a tournament. If you win 30 chips, then lose 30 chips, how many chips do you have? Obviously the answer is 500.

If you remember from part 6, we can do whatever we want to equations as long as we do the same thing to both sides. This single rule combined with a small amount of logical thinking allows us to do a whole lot of things. For example, suppose we have the equation

x + 566 = 1027

and we want to figure out what x is. After a bunch of guesses you could probably figure it out eventually, but it would take some time. Instead, let's use a little logic. What if you subtracted 566 from both sides?

x + 566 - 566 = 1027 - 566

We've stuck to our rule from part 6, so we know this equation has to still be true. But look at what's happened to the left side of the equation. If we start with x, add 566, but then subtract 566, we have what we started with.

x = 1027 - 566

A quick bit of subtraction tells us that x = 461. If we check our original equation x + 566 = 1027, then substitute in 461 for x, we have 461 + 566 = 1027. A quick check with a calculator shows that this is correct, so we have found the right number for x.

It's this process of using logic plus our rule about doing the same things to both sides of equations that will allow us to do really powerful things in poker math.

Poker Example 6: At PokerStars, some player makes 0.27 FPPs per hand at their game. How many hands will it take them to get 50000 FPPs (for the PlatinumStar bonus)?

First let's set up an equation. Let's call the number of hands our player has played H. Then the number of FPPs our player will make over H hands is 0.27H. We want to know how many hands it takes to get 50000, so we'll set 0.27H equal to 50000. That gives us the equation

0.27H = 50000

From here we can divide both sides by 0.27, giving us

0.27H/0.27 = 50000/0.27

The left side of the equation just becomes H since we multiplied it by 0.27 then divided it by 0.27 and those are opposite operations. That leaves us with

H = 50000/0.27

and by using a calculator we see that 50000/0.27 = 185185. So it will take about 185185 hands for our Hero to get his $650 bonus.

Probability Section

Introduction

Probability is basically the study of how often stuff happens. We need to understand some basic concepts about probability before we can start working on tougher EV calculations. This section is simpler than the two before it because the ideas needed are really basic.

Part 9: Outcomes

Before we can get into actual EV calculations, you have to be able to correctly determine all of the possible outcomes. Outcomes are the different ways a situation can play out.

Poker Example 7: Suppose we go all-in on the flop against one player on a semi-bluff, how many outcomes are there?

1. Our opponent folds
2. Our opponent calls, we win
3. Our opponent calls, we lose

Side note: Technically there is a 4th possible outcome in some situations. Our opponent could call and we could tie. We aren't worried about this in the vast majority of situations since equity calculations take into account the ties and split them evenly among the players left in the hand.

Part 10: Finding the Chance of an Individual Outcome

Suppose we have two independent events. (Independent events are two events that don't have any influence on each other whatsoever.) Let's say our chance of winning the first event is 30% and our chance of winning the second event is 50%. Then what's our chance of winning both events? The way to get this answer is to multiply the chance of winning the first event (0.30) times the chance of winning the second event (0.50). So we have 0.30 * 0.50 = 0.15, or 15%.

Poker Example 8: If we go all-in with AA pre-flop against a random hand, we have 84.93% chance to win (pure chance to win, not tie) according to PokerStove. What's the chance of going all-in with AA five hands in a row heads-up and winning all 5?

The chance to win the first time is 84.93%, the chance to win the second time is 84.93%, and so on. So the chance to win all five is 0.8493 * 0.8493 * 0.8493 * 0.8493 * 0.8493 = 0.4419 = 44.19%.

Part 11: The Individual Chances of Each Outcome Must Add Up to 100%

This is a simple idea, which is why it's coming first. Let's say in some situation there are three possible outcomes. The chance of outcome #1 happening is 50% and the chance of outcome #2 happening is 20%. What's the chance of outcome #3 happening? The answer is obviously 30%.

While this seems obvious, you'd be surprised by the number of times a long-term winning small or mid-stakes player has asked me about some EV equation they're working on when the sum of the chances of the outcomes don't add up to 100%.

If you add up all of the chances of each individual outcome and don't come up with 100%, you probably didn't list all of the outcomes.

EV Calculation Section

Introduction

Now we're to the fun part. Instead of trying to analyze every kind of spot, I'm going to work out a few examples in this section to show how the general process works so hopefully you'll have enough to go on that you can do your own calculations yourself.

I'll make one note here about something so people don't get confused. It's often useful in big bet games like NLHE or PLO to let the pot equal 1 and then to call our bet size B. The reason this is useful is because then the value B tells us the % of the pot being bet. For example, if the pot is 1 and B is 0.5, then we know we've bet one-half of the pot. If B was 2, we know we're betting twice the pot, and so on. It's just another little thing to make our life easier to and give us one less variable to have to move around, even though I'm not going to use it here.

Part 12: The EV of Individual Outcomes and Total EV

The EV of an outcome is the chance that outcome happens times your profit when it does. For example, if we're flipping a fair coin, and the times it comes heads you gain $3, then the EV you get from the times you hit heads is 0.50 * $3 = $1.50. If the times it comes tails you lose $1, then the EV you get from the times you hit tails is 0.50 * -$1 = -$0.50.

If you add up the EV of every possible outcome in a situation, then you get the total EV of that situation. So in the above situation since there are only two outcomes, your EV of flipping the coin is $1.50 - $0.50 = $1.

Part 13: Basic Semi-Bluff Shove Calculation

Now it's time for some poker. Suppose we're heads-up on some street and our opponent has us covered. The pot size is P and we're considering making a shove of size B. Our opponent's chance of folding is F, and when he calls our equity is E. Let's find the EV of this shove.

First we have to figure out all of the possible outcomes. The first decision that happens after we shove is that either Villain folds or Villain calls. If Villain folds the hand is over.

Outcome #1: Villain folds

If Villain calls, either we win the hand or we lose. (Note again we're ignoring ties since that's taken care of in our equity %). Both of these outcomes end the hand.

Outcome #2: Villain calls, we win
Outcome #3: Villain calls, we lose

Now we have to find the chance of each outcome happening. The chance of outcome #1 is just the chance of Villain folding, and we know that's F. The chance of outcome #2 is the chance of Villain calling times the chance we win the hand after he calls (our equity). The chance Villain calls is (1-F) which just means 100% minus the % of the time he folds, and our equity is E, so the chance of outcome #2 is E(1-F), and remember that means E TIMES (1-F). The chance of outcome #3 is the chance Villain calls times the chance we lose after he calls. The chance Villain calls is (1-F) like before, but the chance we lose after Villain calls is (1-E), or 100% minus our equity. So the chance of outcome #3 is (1-F)(1-E). In summary, the chances of each:

Outcome #1, Villain folds: F
Outcome #2, Villain calls, we win: E(1-F)
Outcome #3, Villain calls, we lose: (1-F)(1-E)

(Side note: For this next part, it might be useful for those not super familiar with Algebra to know that (a+b)(c+d) = ac + ad + bc + bd.)

So we need to first make sure that all of these chances added together equal 1. So we have:

F + E(1-F) + (1-F)(1-E)
= F + E - EF + 1 - E - F + EF
= 1

Now that we've verified that we have all of the outcomes and they all add up to 100%, we need to find the EV of each individual outcome. Recall from part 12 that the EV of an outcome is the chance it happens multiplied by the profit we get when it happens. In outcome #1, our profit when Villain folds is the size of the pot which is just P. So our EV from outcome #1 is PF, or P times F. In outcome #2, our profit when Villain calls and we win is the size of the pot P plus the size of the bet he calls B, so our total profit in this outcome is (P+B). Since the chance of outcome #2 happening is E(1-F), the EV of outcome #2 is E(1-F)(P+B). Finally in outcome #3, our profit when Villain calls and we lose the hand is a loss of our bet size B, so our profit in this outcome is -B. Since the chance of outcome #3 happening is (1-F)(1-E), our EV is (1-F)(1-E)(-B). In summary, the EV of each:

Outcome #1, Villain folds: PF
Outcome #2, Villain calls, we win: E(1-F)(P+B)
Outcome #3, Villain calls, we lose: (1-F)(1-E)(-B)

Remember that the total EV of this situation is the EV of each individual outcome. Therefore:

EV = PF + E(1-F)(P+B) + (1-F)(1-E)(-B)

TA DA. Now you can put this in a spreadsheet or something and play around with different values to see how different semi-bluffs work out.

Part 14: Continuation Betting a Gut-shot

I'm going to hold your hand a little less on this one. At NLHE with $2/4 blinds and infinite stacks, you’re out of position heads-up on the flop and a $30 pot before you act. You have a gutshot (4 outs) and no other equity. Your gutshot is the nuts and if you hit it on the turn you will win 100% of the time by showdown (if you don’t hit it then you never win the pot other than making him fold on the flop). You c-bet the flop for $20. He will raise 1/3 of the time that he continues to your c-bet (and you must fold), and the other 2/3 of the time he just calls. If you hit on the turn you will bet $60 into the pot of $70. He will call this bet 25% of the time, and 25% of the time he calls the turn bet he will also call a $120 bet on the river. How often does he have to fold to your flop c-bet to make this profitable?

With EV calculations that have a large tree, one way to keep it organized is to list the outcomes and the EV of them individually. This helps to make sure that we don’t miss any possible outcomes, and it also makes it easier if we want to manipulate some variables later on. So here we have a number of possible outcomes:

1. He folds the flop: we’ll call his fold % x, then the EV of this case = 30x
2. He raises the flop: he does this 1/3 of (1-x) % of the time and we lose $20, so the EV = (1/3)(1-x)(-20)
3. He calls the flop and we miss: he calls 2/3 of (1-x) % of the time and we miss 43/47 of the time and we lose $20, so the EV = (2/3)(1-x)(43/47)(-20)
4. He calls the flop and we hit and he folds to a turn bet: he calls the flop 2/3 of (1-x) % of the time, we hit 4/47 of the time, he folds the turn 3/4 of the time and we win the $30 from the flop pot and the $20 c-bet he called, so the EV = (2/3)(1-x)(4/47)(3/4)(50)
5. He calls the flop, we hit, he calls the turn, folds the river: he calls the flop 2/3 of (1-x) % of the time, we hit on the turn 4/47 of the time, he calls the turn 1/4 of the time, he folds the river 3/4 of the time, and we win the $30 flop pot plus the $20 c-bet and $60 turn bet, so the EV = (2/3)(1-x)(4/47)(1/4)(3/4)(110)
6. He calls the flop, we hit, he calls the turn, calls the river: he calls the flop 2/3 of (1-x) % of the time, we hit on the turn 4/47 of the time, he calls the turn 1/4 of the time, he calls the river 1/4 of the time, and we win the $30 flop pot plus the $20 c-bet he calls, the $60 turn bet and the $150 river bet, so the EV = (2/3)(1-x)(4/47)(1/4)(1/4)(260)

So now we have a big ass equation of one variable, 0 = 30x + (1/3)(1-x)(-20) + (2/3)(1-x)(43/47)(-20) + (2/3)(1-x)(4/47)(3/4)(50) + (2/3)(1-x)(4/47)(1/4)(3/4)(110) + (2/3)(1-x)(4/47)(1/4)(1/4)(260). I plug that into something like the equation solver at Algebra Simplifier and Math Solver and get x = 0.328, though you're welcome to do it by hand if you want. So Villain has to fold about 32.8% of the time.

Edit: There are more parts later in this thread at this link spoonitnow's Mathematics of EV Thread

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dranger7070

05-21-2010, 04:35 AM

#2 (permalink)

4-of-a-Kind

Join Date: Oct 2008

Location: Wake up in the mornin feelin' like P. Diddy

Posts: 2,524

dranger7070 has a little shameless behaviour in the past

*head asplode*

nh spoon

Udyr

05-21-2010, 05:15 AM

#3 (permalink)

High Card

Join Date: May 2010

Posts: 10

Good examples.

I definitely appreciate the math behind all of this; most of it can be done on the fly.

I would highly recommend working these out for yourself if you are rusty on the math skills.

EasyPoker

05-21-2010, 08:56 AM

#4 (permalink)

Full House

Join Date: Nov 2009

Posts: 753

[20:19] <Zill4> god
[20:19] <Zill4> u guys
[20:19] <Zill4> so fking hopeless
[20:19] <Zill4> and dumb

Robb

05-21-2010, 10:36 AM

#5 (permalink)

4-of-a-Kind

Join Date: Aug 2007

Posts: 3,072

Robb is an unknown quantity at this point

nh spoon

My Operation and FTR Rethread: Stations are *sob* so hard to play against

tomato paste carnage

05-21-2010, 04:47 PM

#6 (permalink)

Straight

Join Date: Sep 2009

Location: in a van down by the river

Posts: 115

Nice to see this all in one place. Thanks spoon

Tilt is poker cancer. You catch it, you die.

HarleyGuy13

05-22-2010, 01:24 AM

#7 (permalink)

Full House

Join Date: Sep 2008

Location: Oregon

Posts: 1,226

SPOON IS A FUCKING GIVER! Dude seriously I haven't worked on this yet cuz I'm at work but just glancing through it all I can say is wow. You really do a lot buddy and it's appreciated. I know I am one of the ones you are probably tired of having to answer questions for!

"You start the game with a full pot o’ luck and an empty pot o’ experience...
The object is to fill the pot of experience before you empty the pot of luck."

Quote:

Originally Posted by XxStacksxX

Do you have testicles? If so, learn to bet like it

EasyPoker

05-22-2010, 01:45 AM

#8 (permalink)

Full House

Join Date: Nov 2009

Posts: 753

Spoon is a legend

[20:19] <Zill4> god
[20:19] <Zill4> u guys
[20:19] <Zill4> so fking hopeless
[20:19] <Zill4> and dumb

xpaand

05-22-2010, 03:05 AM

#9 (permalink)

Flush

Join Date: Mar 2009

Location: Vancouver

Posts: 289

Spoon, I don't need to tell you anymore about how I feel about you or your ability to teach. Just want to say thank you and that this forum would not be the same without you. Very nice hand sir.

OP: Beginner to Master

If I bet as a bluff, I should be thinking "am I getting better hands to fold? Is it likely that he will fold x% of the time to a y sized bet to make it +EV?". If I bet for value, I should be thinking "am I getting worst hands to call? Am I ahead of enough of his range that this is a good value bet?".

spoonitnow

05-22-2010, 12:01 PM

#10 (permalink)

Straight Flush

Join Date: Sep 2005

Location: IRC Chat Room

Posts: 5,406

I tried to add this to the OP but got the message, "The text that you have entered is too long (27388 characters). Please shorten it to 25000 characters long." Whoops.

Part 15: Break Even Fold Percentage for a Pure Bluff

Suppose we're heads-up against a single opponent. We have a pot size of P and we're considering making a bet size of B as a pure bluff, meaning we have no equity when called. Our opponent will fold F % of the time. There are two possible outcomes. In the first outcome, if he folds F% of the time, we win the pot P. In the second outcome, if he doesn't fold (1-F)% of the time, we lose our bet size B. Therefore, our EV equation is

EV = PF + (1-F)(-B)

Now let's suppose that we wanted to look when this pure bluff is break even. A break even play by definition has an EV of 0, so we let EV be 0 in our equation.

0 = PF + (1-F)(-B)

Now we can solve for F to find what fold % we need to have to make our pure bluff break even. From there, we'll know that if we can make our opponent fold more often than whatever that break even value is that our pure bluff will be +EV. So we have

0 = PF + (1-F)(-B)
0 = PF - B + BF
0 = PF + BF - B
0 = F(P + B) - B
B = F(P + B)
B/(P+B) = F

This shows that our break even fold percentage for a pure bluff is bet/(bet+pot), something a lot of you probably already knew.

Part 16: Break Even Fold Percentage for a Semi-Bluff

In part 13 we found that our equation for the EV of a semi-bluff shove is

EV = PF + E(1-F)(P+B) + (1-F)(1-E)(-B)

where P is the pot size, B is our bet size, F is how often our opponent folds, and E is how much equity we have against our opponent's calling range. If we let EV be 0 like in part 15, we can solve for our break even fold percentage in the same way, though it's a little more difficult, and a little more difficult to interpret the results. So here we go:

0 = PF + E(1-F)(P+B) + (1-F)(1-E)(-B)
0 = FP + E(P + B - FP - BF) + (1 - E - F + EF)(-B)
0 = FP + EP + BE - EFP - BEF - B + BE + BF - BEF
0 = FP + EP + 2BE - EFP - 2BEF - B + BF
0 = BF - EFP - 2BEF + FP - B + EP + 2BE
0 = BF - EFP - 2BEF + FP - B + E(P + 2B)
B - E(P + 2B) = BF - EFP - 2BEF + FP
B - E(P + 2B) = BF - EF(P + 2B) + FP
B - E(P + 2B) = F(B - E(P + 2B) + P)
(B - E(P + 2B))/(B - E(P + 2B) + P) = F

When our opponent’s fold % is greater than that big glob of stuff at the end (which we’ll simplify immensely in a moment), our semi-bluff is +EV. So now we have the task of figuring out how to do something with this. If we let B – E(P+2B) be an entity of its own (call it x), then we’d have F > x/(x+P). This is exactly like the equation we solved for earlier when looking at a pure bluff! Now let’s figure out what B – E(P+2B) means since it seems like it might be important. If you notice, (P+2B) is the total size of the pot after we bet and Villain calls. Then, E(P+2B) is our equity % of that total pot size. Finally, B – E(P+2B) is our bet size minus our equity % of the total pot size. So let’s see this in an example.

Say it’s 25nl and the pot is $30 and we have $20 behind heads-up. If we shove our $20 and we think we’ll have 20% equity when we’re called, how often does Villain have to fold for it to be +EV? First, we’ll find the total pot size after Villain calls, and that’s $70. Second, we’ll find our equity’s percent of that pot, and 20% of $70 is $14. Now we subtract $14 from our bet size of $20 to get $6, which is our x value. To finish, our Villain’s fold % has to be greater than x/(x+P) for our semi-bluff to be +EV, which is 6/(6+30) here, or 1/6 = 16.7%.

sven00100

05-23-2010, 06:03 AM

#11 (permalink)

3-of-a-Kind

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Posts: 84

Quote:

Originally Posted by spoonitnow

So now we have a big ass equation of one variable, 0 = 30x + (1/3)(1-x)(-20) + (2/3)(1-x)(43/47)(-20) + (2/3)(1-x)(4/47)(3/4)(50) + (2/3)(1-x)(4/47)(1/4)(3/4)(110) + (2/3)(1-x)(4/47)(1/4)(1/4)(260). I plug that into something like the equation solver at Algebra Simplifier and Math Solver and get x = 0.328, though you're welcome to do it by hand if you want. So Villain has to fold about 32.8% of the time.

I got bored. so i took 2 minutes to do this haha..

0 = 30x + (1/3)(1-x)(-20) + (2/3)(1-x)(43/47)(-20) + (2/3)(1-x)(4/47)(3/4)(50) + (2/3)(1-x)(4/47)(1/4)(3/4)(110) + (2/3)(1-x)(4/47)(1/4)(1/4)(260)

So.. first we multiply out all of the parenthesis..

0 = 30x – 20/3 + 20x/3 – 1720/141 + 1720x/141 + 100/47 – 100x/47 + 55/47 – 55x/47 + 130/141 – 130x/141

Then we move all of the characters which don't contain x to the opposite side of the equation.

20/3 + 1720/141 – 100/47 – 55/47 – 130/141 = (30)x + (20/3)x + (1720/141)x – (100/47)x – (55/47)x – (130/141)x

Knowing that 141 is the multiple of prime numbers 3*47; we know the LCD (lowest common denominator) of these characters is 141. so each term is multiplied by 1 in the form of whichever it lacks ( ie. 20/3 is multiplied by 47/47 to get (20*47)/(3*47) = 940/141 )

940/141 + 1720/141 – 300/141 – 165/141 – 130/141 = x(4230/141 + 940/141 + 1720/141 – 300/141 – 165/141 – 130/141)

Then we can simply add together all of the numerators, as they have a common denominator:

2065/141 = x(6295/141)

Multiply each side by the reciprocal of the coefficient attached to x:

x = (2065/141)(141/6295)

And voila!

x = (2065/6295) = .3280381254964257…

daven

05-23-2010, 06:38 AM

#12 (permalink)

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i value the stuff you have posted on FTR. This one is less valuable for me, but it's gold for those who need it! very nice post.

-> beginners circle. If you are scared of the math, don't be. Work your way through this post, one section at a time. If you get stuck then jump into IRC, you'll get help. But try first, this is very well presented and there is a good chance you'll manage to get through it.

don't raise with the straight flush bro, nobody ever gonna call that shit
allin, reload, repeat

please read: http://www.philgalfond.com/lets-make-some-changes/

xpaand

05-23-2010, 07:02 AM

#13 (permalink)

Flush

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Quote:

Originally Posted by daven

Same here, but you can't deny the effort. Wow. And your last note is so important imo. BC'ers, don't let it scare you and maybe more importantly, don't disregard it as too basic without looking over it. Maybe you know the mathematics part but not the poker applications and the way you can analyze many many many situations mathematically. Had a tough time with the more complex applications of expected values last year when I hadn't done any math in years.

rpm

06-01-2010, 11:39 AM

#14 (permalink)

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just got through the first chunk. i'll probably re-read it and try a few examples of my own to let it sink in before i move on. being one of the BC'ers who has asked you for help with poker-related EV equations in the past, seeing it all in the one spot and especially with a refresher course on the required background concepts makes a hell of a difference. thanks alot for this spoon.

out of interest, through doing this sort of stuff extensively in your poker analysis, and probably in your schoolwork as well, i'm sure the EV equation is heavily incorporated into your thought process during the play of a hand.... is this through intuition (ie you develop a feel for +EV plays based on the differing information presented to you in the hand - potsize, betsize, effective stacks, opponents "F:C Ratio", equity versus calling range etc), or through precise mental calculation?

cliche question i'm sure, but i'm fascinated to know, because i'm not mathematically inclined, and i'm wary of attempting to adopt TOO mathematical a mindset during play because i feel it could potentially just cloud my thinking as i inevitably fumble mentally with all the numbers and equations etc etc in my race against the timebank.
thanks again.

Santo2True

06-01-2010, 12:25 PM

#15 (permalink)

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great post spoon. gone through it several times and will be referring to it very often

"Those who say it can't be done, shouldn't interrupt those who are doing it"

spoonitnow

06-01-2010, 04:51 PM

#16 (permalink)

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Quote:

Originally Posted by rpm

is this through intuition (ie you develop a feel for +EV plays based on the differing information presented to you in the hand - potsize, betsize, effective stacks, opponents "F:C Ratio", equity versus calling range etc), or through precise mental calculation?

Calculations in your study help you develop the intuition so you can feel your way around when you're playing.

Quote:

Originally Posted by Ripptyde

I only have 2 simple rules when I am coaching a new student.

Rule # 1: don't ask questions

Rule # 2: don't ask questions

I have no interest in discussing strategy with a protege'. Your job is to remain quiet and listen. I have a very systematic approach that I will share with the right candidate and I promise that I will turn you into a force of nature and show you elements of the game of poker that you never knew existed.

donkfish

06-01-2010, 09:17 PM

#17 (permalink)

3-of-a-Kind

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Thanks for all the math posts dude. That's where I feel my game is most lacking. I plan on reading through the entire collection (this thread + the ones in the digest) over the next few days and really buckle down.

littleogre

06-01-2010, 11:38 PM

#18 (permalink)

Join Date: Oct 2004

Posts: 1,344

littleogre has a little shameless behaviour in the past

Spoon do you take request? If so what would be the formula for solving the following situation? We are in a 2nl game hu ip. The villain just bet .75 into a 1$ pot on the turn we raise 1.75 with a nut flush draw+gs nut str8 draw that will win 100 percent of the time when it hits . Money behind=1.75if he just calls if he raises 3-bets it will be a shove so either way we stand to win a total pot of 4.25. What is the formula for figuring this out? I think it would be similar to #14 from your list but likely a little different as we will be calling his turn shove due to odds.

spoonitnow

06-01-2010, 11:57 PM

#19 (permalink)

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Quote:

Originally Posted by littleogre

Figure out each possible outcome, find the chance of each outcome, find your profit for each outcome, etc. just like in the examples.

littleogre

06-02-2010, 02:00 AM

#20 (permalink)

Join Date: Oct 2004

Posts: 1,344

Quote:

Originally Posted by spoonitnow

Figure out each possible outcome, find the chance of each outcome, find your profit for each outcome, etc. just like in the examples.

Ok well i suck at structuring algebraic formulas but i'll try after i eat dinner.

littleogre

06-02-2010, 02:31 AM

#21 (permalink)

Join Date: Oct 2004

Posts: 1,344

ok well it's gonna be a few before dinner is ready. I will be use percents instead of fractions. ok i think i left out some info in the first post

pot size=1.75
bet=1.75
p+b=3.5
his fold%=30
his call&=40
his raise%=30
our chance oof hiting our draw aprox .28
his fold% to river shove=60
his call percentage=40
he will always shove if he 3-bets the turn
ev of him folding to our turn bluff .30*1.75 =0.525
ev of him calling and us missing .40*(-1.75) =-.7
ev of him calling us hittin and him folding= ( this is where i start getting lost)
.28(3.5)(.60)=.588

ok i'll continue later

Tasha

06-02-2010, 02:41 PM

#22 (permalink)

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Many thanks, this has clarified a lot. However, I have one question: In section 14, the 'big ass' equation at the end equals 0 because we are looking for the minimum value that will give EV+, is that right?

spoonitnow

06-02-2010, 04:24 PM

#23 (permalink)

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Quote:

Originally Posted by Tasha

It equals zero because we're looking for how often he has to fold for it to be =EV. The reason we look for that value is that we'll know if he folds more often than whatever that value is then it will be +EV.

littleogre

06-02-2010, 08:19 PM

#24 (permalink)

Join Date: Oct 2004

Posts: 1,344

lets continue
pot size=1.75
bet=1.75
p+b=3.5
his fold%=30
his call&=40
his raise%=30
our chance oof hiting our draw aprox .28
his fold% to river shove=60
his call percentage=40
he will always shove if he 3-bets the turn
ev of him folding to our turn bluff .30*1.75 =0.525
ev of him calling and us missing .40*(-1.75) =-.7
ev of him calling us hittin and him folding= ( this is where i start getting lost)
.28(1.75)(.60)=.588=.29
ev of him calling us hiting and him paying off
ev=.28(3.5)(.40)=.39
ev of him raising and us missing .72(-3.5)=-2.52
ev of him raising and us hitting =.28(7)=1.96
tot ev =-.55

littleogre

06-02-2010, 08:37 PM

#25 (permalink)

Join Date: Oct 2004

Posts: 1,344

so i think the formula would look like this
ev=.35*.75+.40*-1.75+.28*1.75*.6*+.28*3.5*.40+.72*-3.5+.28*7

ok for some reason i keep getting invalid expression

spoonitnow

06-02-2010, 08:45 PM

#26 (permalink)

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I think it's because you didn't have your negatives in parenthesis and near the middle you have a stray * between .6 and .28. Here's the output from the PowerToy Calc after I fixed those two things:

.35 * .75 + .40 * (-1.75) + .28 * 1.75 * .6 + .28 * 3.5 * .40 + .72 * (-3.5) + .28 * 7
-0.3115

littleogre

06-02-2010, 09:10 PM

#27 (permalink)

Join Date: Oct 2004

Posts: 1,344

Quote:

Originally Posted by spoonitnow

Except for my 2 over sites the formula is correct ?

spoonitnow

06-02-2010, 10:20 PM

#28 (permalink)

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Quote:

Originally Posted by littleogre

Except for my 2 over sites the formula is correct ?

Idk I couldn't really understand the situation you were talking about. You didn't really clearly state all of the possible outcomes.

littleogre

06-03-2010, 12:33 AM

#29 (permalink)

Join Date: Oct 2004

Posts: 1,344

Quote:

Originally Posted by spoonitnow

Idk I couldn't really understand the situation you were talking about. You didn't really clearly state all of the possible outcomes.

yes clarity is sometimes a problem for me but i'll try again later.

rpm

06-03-2010, 09:53 AM

#30 (permalink)

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hypothetical scenario to help heal my confusion. pot is 50 (precisely 25 each went in pre) on the flop and villain leads into us for 25 (leaving him 50 behind) and we decide to raise with our last 75. ignoring our equity and villains fold frequencies because they're irrelevant to my question:

if we set P to 75 (ie how much we stand to win when he folds), then P+B is not what we win when he calls, because villain only has 50 behind, and his 25 bet is already counted in the P value. so we stand to win P+effective stack. do we just make another variable to correlate to "effective stack" and use that for what we stand to win?

now i have typed this out, i think i have answered my own question but i'll post it anyway for clarification. it seems that it's just a difference in equation between semi-bluff betting and semi-bluff raising. in your example (spoon) we were betting, so assuming we have identical stack size to villain, we win the pot + the size of our bet when called. whereas when we are semi-bluff raising, we win the pot + the effective stack. am i correct here at all?

spoonitnow

06-03-2010, 04:46 PM

#31 (permalink)

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Yes because he's going to be calling a smaller amount than we're betting, etc.

rpm

06-04-2010, 02:22 AM

#32 (permalink)

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easy like sunday morning

spoonitnow

06-06-2010, 10:06 PM

#33 (permalink)

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Part 17: Existence of the Positional Advantage

We all know that the player who is last to act has an advantage because of position. There are various things that contribute to this advantage, like information, but those specifics are outside of the scope of this thread.

In the book "The Mathematics of Poker", the authors find the unexploitable strategies for various toy poker games, including heads-up push/fold no-limit hold'em. In each of the AKQ-style toy games (fixed-limit, spread-limit, no-limit) it's the player with position that has an advantage when the starting ranges between each of the players is the same and each player is playing unexploitably.

player in position can force an advantage. Intuitively, this seems obvious, but it's nice seeing some solid evidence. So we can speculate that the following theorem is true:

Positional Theorem: If all players in a hand have the same range, the player in position can force an advantage if all else is equal.

rpm

06-26-2010, 05:17 AM

#34 (permalink)

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digest-worthy imo

spoonitnow

07-08-2010, 05:32 PM

#35 (permalink)

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Bump.

spoonitnow

08-11-2010, 07:51 PM

#36 (permalink)

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Bump.

HarleyGuy13

10-23-2010, 01:23 AM

#37 (permalink)

Full House

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bump

unclesteve

10-23-2010, 02:47 PM

#38 (permalink)

3-of-a-Kind

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Holy shit.

Tasha

10-25-2010, 11:42 AM

#39 (permalink)

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This was one of the first threads I ever read here. If you haven't read it yet then stop what you are doing and start reading it now.

spoonitnow

11-13-2010, 04:02 AM

#40 (permalink)

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Part 18: Side Pots

This gets brought up from time to time, so I thought I'd make a short post about the basic way to work this out as an example here in this mathematics of EV thread.

Example 1: Suppose we go all-in preflop 3-way. We have 100bb with AKo, one Villain has 100bb with {99+, AQ+}, and a third Villain has 30bb with {JJ+, AK}. There are two pots. In the main pot, we are 30bb deep three-way, and in the side pot we are playing for another 70bb against only one player. So we add the EVs of the two pots together to get our total EV. Here are our equities from PokerStove:

Code:

	equity 	win 	tie 	      pots won 	pots tied	
Hand 0: 	24.002%  	15.55% 	08.45% 	    2777357652 	1510288016.00   { AKo }
Hand 1: 	33.593%  	30.31% 	03.28% 	    5414962260 	585995198.00   { 99+, AQs+, AQo+ }
Hand 2: 	42.405%  	35.43% 	06.98% 	    6328382976 	1246680026.00   { JJ+, AKs, AKo }

	equity 	win 	tie 	      pots won 	pots tied	
Hand 0: 	48.652%  	39.52% 	09.13% 	     414180516 	 95654640.00   { AKo }
Hand 1: 	51.348%  	42.22% 	09.13% 	     442440252 	 95654640.00   { 99+, AQs+, AQo+ }

EV of main pot = 0.24002(60) + 0.75998(-30) = -8.3982
EV of side pot = 0.48652(70) + 0.51348(-70) = -1.8872

So our total EV is (-8.3982) + (-1.8872) = -10.2854 big blinds.

Example 2: We're 4-way and we're all-in with cards left to come. The stacks are 120bb (us), 120bb (Villain A), 75bb (Villain B), 12bb (Villain C). There are three pots:

Main Pot - We're betting 12bb four-way.
Side Pot 1 - We're betting another 63bb three-way.
Side Pot 2 - We're betting another 45bb two-way.

Let's say for the sake of example and nice, round, even numbers that we have 27% equity in the main pot, 31% equity in side pot 1, and 35% equity in side pot 2. We need to find our EV in each pot, and add them together to find our total EV.

EV of Main Pot = 0.27(36) + 0.73(-12) = 0.96
EV of Side Pot 1 = 0.31(126) + 0.69(-63) = -4.41
EV of Side Pot 2 = 0.35(45) + 0.65(-45) = -13.5

And the total EV is 0.96 - 4.41 - 13.5 = -16.95.

spoonitnow

11-13-2010, 06:31 AM

#41 (permalink)

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Part 19: Calling to Close the Action in Split Pot Games

Warning: This is math-intensive. Feel free to skip if you don't play split games.

Okay this is pretty cool and I'm going to pretty much just be posting this because I think it's neat. The general idea of split-pot games is that we have two equities, one for the high pot and one for the low pot, where the high pot and low pot are half of the total pot each. In games with a qualifier for the low, sometimes there won't be a low pot, which adds another layer of complexity to the EV calculation.

So let's suppose we're playing a game where there is a high/low split with no qualifier, someone shoves and we call. Let's call our high pot equity 'a', and call our low pot equity 'b'. Also, we're calling a bet of 1 with a pot of P when it's our turn to act. To find the EV of this call, first let's find the possible outcomes and our profit for each.

Outcome 1: We win both pots. The chance of this happening is ab. We will profit P.
Outcome 2: We win the high pot and lose the low pot. The chance of this happening is a(1-b). The profit here is a little tricky. Suppose the pot was 5 bb and we call our 1 bb, then we get 3 bb back. Our profit is (5+1)/2 - 1. So our profit is (P+1)/2 - 1.
Outcome 3: We win the low pot and lose the high pot. Same profit as outcome 2, but the chance of it happening is b(1-a).
Outcome 4: We lose both pots. The chance of this happening is (1-a)(1-b) and our profit is -1.

As always, we'll add all of these possible outcomes together to get our EV:

EV = abP + a(1-b)((P+1)/2 - 1) + (a-1)(b)((P+1)/2 - 1) + (1-a)(1-b)(-1)

This is a bit hairy because of that (P+1)/2 - 1 bit. Let's break it down to something easier to work with:

(P+1)/2 - 1
(P+1)/2 - 2/2
(P+1-2)/2
(P-1)/2

If we do the substitution we get:

EV = abP + a(1-b)((P-1)/2) + (1-a)(b)((P-1)/2) + (1-a)(1-b)(-1)

Which is still a bit hairy, but we can do it:

EV = abP + a(1-b)((P-1)/2) + (1-a)(b)((P-1)/2) + (1-a)(1-b)(-1)
EV = abP + a(1-b)(P-1)/2 + (b)(1-a)(P-1)/2 + (1-a)(1-b)(-1)
EV = abP + a(P - 1 - bP + b)/2 + b(P - 1 - aP + a)/2 + (-1)(1 - a - b + ab)
EV = abP + aP/2 - a/2 - abP/2 + ab/2 + bP/2 - b/2 - abP/2 + ab/2 - 1 + a + b - ab

Let's look at the break even point (i.e.: when EV=0).

0 = abP + aP/2 - a/2 - abP/2 + ab/2 + bP/2 - b/2 - abP/2 + ab/2 - 1 + a + b - ab
0 = abP + aP/2 - abP/2 + bP/2 - abP/2 - a/2 + ab/2 - b/2 + ab/2 - 1 + a + b - ab
0 = aP/2 + bP/2 - a/2 - b/2 - 1 + a + b
0 = aP/2 + bP/2 + a/2 + b/2 - 1
0 = aP + bP + a + b - 2
0 = P(a+b) + (a+b) - 2
0 = (P+1)(a+b) - 2
2 = (P+1)(a+b)
1/(P+1) = (a+b)/2

Remember that 1/(P+1) is just our standard bet/(bet+pot). Also notice that (a+b)/2 is just our average equity between the two. Therefore, when our average equity is greater than bet/(bet+pot), then we should call. (Notice also that this is a more general form of our earlier equation for finding the break even point for calling in a game without a split pot.)

spoonitnow

12-08-2010, 06:47 AM

#42 (permalink)

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Bump.

A link to a new thread probably belongs in this one for future reference:

http://www.flopturnriver.com/pokerfo...me-184365.html

daviddem

02-15-2011, 08:14 PM

#43 (permalink)

Full House

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Bump, and could one of the mods please link to this in the beginner's digest? Don't understand why it's not in there.

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