Texas Holdem Equity
Texas Hold'em Equity Calculator
I am wondering how to mathematically calculate equities in equity based poker variants( holdem, omaha, Stud ect). Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In Texas Hold'em, poker odds are THE probability tool you need as a poker player. In fact, you should always be thinking about poker odds - yours and your opponents' - when making decisions. In short, poker odds is the probability of you winning that hand, or the price it offers (pot odds). An explanation of the term 'equity' as it is used in no-limit hold'em when estimating the current value of one's hand. + Get the Ultimate Beginner’s Guide to Texas Hold’em - FREE! The Texas Hold’em odds of how likely hands are to unfold after the flop will help guide almost every action you make on the flop Odds On the Flop in Texas Hold’em. The flop is the turning point of a Hold’em hand. This is where you’re going to make your biggest and most expensive decisions. Poker Equity Chart: Common PREFLOP Scenarios. The chart below indicates the preflop equities of various matchups of poker hands when two players see a flop heads-up in Texas Hold’em poker. Regarding preflop equities and the chart above, a few interesting things to note, include the following.
Last week’s article from the “Strategy Vault” shared a past discussion with Andrew “BalugaWhale” Seidman about floating versus bluff-raising after the flop. There Seidman related his thought process when after having called a preflop raise from position, he misses the flop and watches his opponent makes a continuation bet. Folding is always a choice here, but he focuses on instances when he’d rather “float” or call the bet with an intention of bluffing on a later street, or go ahead and bluff right there by raising.
Seidman’s explains how decision in such a spot is affected in part by how his opponent plays, and also by the “equity” he has with his hand that missed the flop. You can go back to that article to see how the combination of his opponent’s style and the relative equity of his hand affects his thinking, but I wanted to stop and linger a little over that term — equity — and talk about how it relates to poker, generally speaking.
What is “Equity”?
The word itself actually means “fairness” — or the quality of being fair or impartial. That idea comes to us sometimes when things don’t seem so fair at the poker table. You get your stack in with versus an opponent’s and a king spikes on the river — not fair!
Equity also refers to someone’s “share” of something, as in shares owned in a company. With your aces you might have claimed a “rightful share” to most of the pot while your opponent with kings has much less of a “right” to what’s in the middle. But after the community cards came and that king fell, you lost your equity and he got every chip.
That situation refers to equity after the all-ins have already come. What about during a hand, when there are still decisions to be made? Of what relevance is equity then?
The first time I ever really bothered to think seriously about what equity meant probably came at some point after buying a home. For those who haven’t gone through the process, most of us don’t buy a home outright, but perhaps make a down payment (some % of the total cost) and take out a mortgage to cover the rest. From there we make payments each month, with the majority of that payment actually going to the interest on the loan and the rest to the principal.
Ultimately your home equity refers to what part of the home you actually “own” as determined by that original down payment plus whatever you’ve paid so far. Also, if the house has gained in value since the time you bought it, that, too, helps build your equity in the home. (By the same token, if your home depreciates in value you lose equity.)
While buying a home can be pretty complicated, the general idea of equity is fairly simple. The equity you have in your home is essentially what you would earn if you were to sell it, something that’s affected greatly by what you “put into” it both at the time you bought it and with the mortgage payments you made afterwards. Before you sell the house you can estimate this figure by looking at the current market value of the house and subtracting your outstanding mortgage balance. If I think I can sell the house for $200,000 and I currently have a $120,000 balance, my home equity is $80,000.
That said, I don’t actually realize that equity until I do sell the house. And the longer I wait, the more things could change between now and then.
Equity = Current Value
The idea of equity works similarly in poker. It essentially represents a theoretical amount you “own” of every pot for which you compete. I say “theoretical” because in the end not everyone gets to realize their equity in every pot — in fact, usually only one person does.
You hold and your opponent has . You raise from late position, he calls from the small blind, building a pot of $300. With five cards to come, you have just under a 45% chance of improving enough to beat pocket sevens. Your equity at this point is 45% of $300, then, or $135.
The flop then comes , giving you top pair. Now with two cards to come, the PokerNews Odds Calculator shows you have a better than 91% chance to win. Suddenly your equity has risen to 91% of $300, or $273. It’s like the market value of your hand just went way, way up. (Unlike what happens when buying a home, equity values change a lot faster in poker hands.) Or you might think of that ace on the flop being a big fat payment lowering the principal of your mortgage. In any case, the future is looking bright.
If you can find a way now to increase the size of the pot, you will also increase your equity. If you bet $100 and your opponent calls, the pot grows to $500 and your equity increases again to $455. For every dollar that goes into the pot here, you increase your equity by 91 cents.
Equity = (Usually) an Estimation
Of course, in an actual poker hand you wouldn’t know your opponent has pocket sevens, so you can’t know exactly how much equity your ace-king gives you. Instead you estimate — top pair, top kicker is likely ahead of most hands your opponent has here, you think, so you probably do have more equity than he does, considering his likely range of hands. That makes it preferable to try to build the pot further and increase that equity.
It was a similar estimation that had inspired you to raise preflop with in the first place. Ace-king is a premium hand, one that stands to be better than most others before the flop, and so your equity would increase by raising and increasing the size of the pot. Home owners looking at a house’s current market value are also making an estimation about how much equity they have in the home, understanding — just like the person with ace-king before the flop does — that it could change in the future.
Going back to Seidman’s discussion, he looks at situations when he missed the flop but may still have equity in the form of a draw that would improve his hand. Thus when he has after a flop he estimates he has practically no equity, while having that same hand on a flop of gives him a flush draw and considerably more equity — a difference (that along with his opponent’s style) will affect his decision whether to call or raise after the continuation bet comes.
Equity does not equal cash in pocket. You still have to close the deal, so to speak, which can be complicated — whether you’re selling a house or trying to win a hand of hold’em. But being able to estimate your equity at any given moment in a hand by accurately comparing its value with the value of others’ hands better informs your decision-making going forward.
Texas Holdem Equity Chart
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Calculating pot odds is a basic skill that most poker players learn soon after they take up the game. Almost all introductory poker books feature a section on pot odds, but they don’t go into too much depth. The truth is that pot odds and equity calculations are at the heart of being a successful poker player and if you wanted to, you could write a book on the subject. In fact, Bill Chen did just that; his book The Mathematics of Poker doesn’t make for light reading though. In this article we’ll go a little beyond the basics of pot odds, but not far enough so that you’ll have to stop and get a PhD in math to continue half way through.
The Basics of Pot Odds
Poker is a game of betting, and all betting revolves around odds. If your opponent bets $100 into a $100 pot, then you have to put in $100 to call. This means you’re risking $100 to win $200, representing odds of 2/1. To figure out if a call is profitable, you need to convert the odds on offer to an implied chance of winning. To do this, you simply add the numerator (above the line, 2 in this case) and the denominator (below the line, 1 in this case) and put the denominator above the sum giving you 1/3, or a 33% chance. Taking another example, let’s say your opponent bets $30 into a $70 pot, meaning you must risk $30 to win $100, so your odds are 100/30. Following the same formula as last time, we end up with 30/130, an implied chance of 23%.
So how does this relate to the cards in your hand? The gravest mistake that bad poker players make is continuing with their hand when the pot odds dictate that they should fold. Again, the easiest way to illustrate this is with an example:
Say you’re playing a tournament and you’re holding 7h8h and there are 5000 chips in the pot on the flop which reads AhTs2h and your opponent goes all in for 5,000 chips. When your opponent bets the size of the pot, we found out already that this gives you odds of 2/1 and you need a 33% chance of winning for you to justify continuing with the hand. If you’re certain than your opponent holds AK in this spot then you know that a flush will beat him if you make it.
If you know how to count outs and how the rule of 2 and 4 works, you’ll know that your flush draw has about a 36% chance of hitting on or before the river. The fact that your chances of winning are greater than the pot odds on offer means that a call will show a long term profit and that you can make it in this spot. In fact you can even calculate how many chips you’ll make from the call on average by adding the average chips gained when you hit, to the chips you lose when you miss. 36% of the time you’ll hit, and win 10,000 chips for a total of 3600, and 64% of the time you’ll lose 5000 chips for a total of -3200. The overall total gives you an expected value of +400 chips on average by making the call.
Let’s imagine the stack sizes are changed and your opponent surprises you by betting all-in for 10,000 chips on the flop. How does that change things? Now you’re risking 10,000 to win 15,000 giving you odds of 15/10 meaning that you need to win [10/(10+15)] = 40% of the time to break even. In this situation if you call, you’ll win 15,000 chips 36% of the time (5400), and you’ll lose 10,000 chips 64% of the time (-6400), so your expected value is now -1000 chips with this call.
In poker, you win when your opponent makes mathematical mistakes. If you bet a tiny amount relative to the size of the pot, you give your opponent very good pot odds and they can very often make a mathematically correct call. If you bet a bigger amount, they’ll generally not be getting the right price to call, and so if they do they’re making a mistake, which means a profit for you in the long run.
Implied Odds
In the examples we’ve discussed so far, the bets we’ve had to call have been all-in bets, and our pot odds calculations have been straightforward. Of course in deeper stacked cash games or early in tournaments, we’ll rarely be facing an all-in bet on the flop. Imagine you’re holding the same 7h 8h on a flop of AhTs2h in a $1/$2 cash game with stacks of $200 and $6 in the pot on the flop. Say your opponent bets $10, giving you odds of 16/10. If his bet was all-in, you’d need a 10/26 = 38% chance of winning the hand to call this bet. Your flush draw doesn’t quite make it, and so if his bet was all-in you should fold. However, with more money to into the pot, you can factor in money you could still possibly win before deciding whether to call or to fold. Let’s say you know this opponent will never fold AK and you’ll stack him if you hit your flush. Now when he bets $10, you’re really being offered his entire stack should you hit, so you need to call the $10 to prospectively win $200 plus the $6 in the pot meaning you’re getting odds of greater than 20/1. In this spot it’s clear that you should continue with your flush draw.
Let’s imagine that the turn comes down the 4d and your opponent bets $20 into the $26 pot, can you still continue. At this stage when implied odds are factored in, you’re being offered the $26 that was initially in the pot, plus his remaining $190 and you need to call $20 to see the river, representing odds of 216/20 or a little better than 10/1, meaning you need to win 9% of the time or more to show a profit.
With just one card to come, you have about an 18% chance of hitting your flush and so you can profitably call again.
Of course this is an extreme example, you can never be 100% sure that you’ll stack your opponent or that you’ll have the winning hand by the river. Even if you make your flush, your opponent could easily fold when you show a lot of strength or show down a higher flush or a full house, which brings us to the subject of reverse implied odds.
Reverese Implied Odds
Our examples above took into account the potential for winning money on later streets if we make our hand. However there are a number of situations in poker where we stand to lose out on money that goes into the pot on future streets. These are known as reverse implied odds situations. An example of reverse implied odds would be calling a re-raise from a tight player with a hand like KQ. If the player has a tight re-raising range then we’ll be put in a lot of difficult spots if we make a hand which we think could be good. For example if he has AK, then we’re in a reverse implied odds spot, as we’re likely to call him down if the flop comes King high, but we’re drawing very thin. Hands where you’re likely to be dominated if you make a pair represent the most common type of reverse implied odds situations.
The most costly scenarios are where you’re drawing to a strong hand but your opponent has a stronger draw, such as a better flush draw or the higher end of a straight. Reverse implied odds situations are particularly difficult to deal with when you’re out of position and will find it difficult to control the size of the pot.