Lesson: EV

Poker involves wagering. Yes, you know it is a form of gambling, in which you put up x dollars as your stake, and the prize is to win y dollars. But every action you take in poker is also a wager.

It's easy to understand what your expectation is for a series of tourneys. Say you play the $10 games with no "rake" (the vig you must pay the poker room for running the table). You have an ROI of 10%. So if you play 100 games, you expect to net (10 x 1.1) x 100 = $1100. Divide that by 100 and you get $11. So your "expected value" for each game is $11. Hang on, you're thinking, you cannot win $11 in an STT with a standard payout scheme. Right. EV does not correspond with immediate reward. It is your average return over the series you're considering. You are going to win some and lose some, but over the whole 100 games, you'll average out to taking $11 a game. In poker, we talk about the long term. This is basically the series of games we will play in our lives, and it doesn't have a term. It's not infinite, but it's not bounded (because we have no idea how many games it will be). So we think about decisions playing out over that long term.

Say you have a flush draw on the turn. Your chances of catching a flush card on the river are a bit more than 4/1. Now, I'm sure boots will tell you that they are in fact either 100% or 0%, and that's true. The next card either is, or is not, a flush card. And even if it is not yet selected, so will be random, some of the flush cards can be in other guys' hands, so your odds are not actually 4/1. None of that matters. You don't know what the next card is and must make a decision. (And actually, there being fewer or more outs actually in existence is balanced out by other factors, and your expectation is not affected by that.)

So your expectation is that you will get your flush one in five times. But you don't know which of the five times you are going to have! What you can do is consider your expectation as 0.2 (which is just another way of writing 20%, because 20% of the time you expect to win the pot), and make your decision accordingly. Note again that in fact you either will or won't make your flush. You will or won't win the pot. But think about that series of tourneys. In them, you either make $40, $20, $10 or $0, but your expectation for each tourney is not one of those figures, but the figure that describes the proportion of each that you expect: $11.

So when you are considering whether to call with a flush draw, you can look at the pot, and look at the bet you face, and if it pays enough, you call the bet. Why? Let's say we are playing limit, where the bets are fixed. The pot has five bets in it and you face paying one bet to see the river. Let's say that if your flush comes, you will not make any more money because your opponent will just fold. So the pot is paying five to one.

Okay. This time, you may win five for your one or you may just lose your bet. But you don't know which it will be. Over the long term, you will win one out of five though, and you will face this bet many times over the long term. What you can do is take the bet when it favours you every time you are offered it. Then, over the long term, you are bound to profit (so long as the laws of probability don't get overturned by the poker gods). A bet that offers better odds than its probability of coming in is said to have "positive expectation" or is "+EV". In this case, you will win 0.2 x 6 = 1.2 bets, and you have to wager one bet. So long as the return is bigger than the amount you're betting, you're +EV.

What if there were only three bets in the pot? Now your expectation is 0.2 x 4 = 0.8 bets. You expect to lose. (If you're wondering why four bets and not three in that equation; it's because when the flush card comes, you win back your bet if you win.) This sort of bet is -EV. Note again, you may win this time. You may win this one and the next one and the next. But if you keep taking -EV bets, you will lose money.

The odds that the pot offers you by virtue of its size are called "pot odds". In most forms of poker, you are willing to take bets where pot odds are greater than the odds of the outcome being favourable. There are three further considerations though that we can mention briefly. (We'll talk about them in more detail later.) First, your opponent does not always fold on the river. Sometimes you will be paid another bet or bets. You can sometimes factor this into your decision whether to call. In our example the pot odds may be 4/1, indicating a fold, but you figure the other guy, fish that he is, will always call on the river. So you can count on another bet, making your "implied odds" 5/1, giving you a call. This is a subtle concept, because the pot odds are certain, but the implied odds vary depending on many factors. Second, you make wagers in tournaments with tournament chips, not with money, but you must consider your bets in terms of dollars. That's not easy but it's essential, because you cannot replace your chips and must weigh their use correctly. Sometimes, chasing that flush, which you would do in a cash game, will be wrong because although it shows a profit in chips, it doesn't show one in dollars. Third, it's straightforward to draw to a flush and win if you make it, lose if you don't. But you are rarely working out your EV so simply. You don't always know what your opponent has, or what he's drawing to. The best you can do is figure out the "range" he has: the hands he is likely to have, and work out your EV against the range.

What do I mean? Well, let's say you have KK. You raise preflop. Some guy calls. He's quite tight, and (I won't go into how) you figure that 55% of the time he has a smaller pair, 40% of the time he has a big ace and 5% of the time AA. The flop comes A73. You are to act. You want to know whether you should bet. You don't know what he actually has though. But you can do what would be right against his whole range. Let's assume you are pushing or folding and he is calling with everything. Let's say the pot has 500 chips in it and you have 1000 more to bet.

He has 77 or 33 10% of the time (a little less because of course there are a 7 and a 3 on the flop, but we aren't looking for precision here). They have a set, which crushes you, and you will "draw out" on them roughly 9% of the time. Let's say you lose 9% of the time to those pairs.

He has AA 5% of the time, and effectively you're "drawing dead" against AA. (You need to turn and river Ks to beat him. Good luck with that.) So you lose 5% of the time to that.

He has a big ace 40% of the time, and that is beating you. About 9% of the time, you'll draw out on those too, so let's say you lose 35% of the time to them.

He has a smaller pair that is not 77/33 45% of the time. They will draw out on you about 9% of the time, so let's say you lose 5% of the time to them.

You're losing 54% of the time. You expect to win less often than you lose, so you fold right?

Wrong. There is money already in the pot. Don't forget the money already in the pot! Your expectation is .46 x 2500 = 1150. So for your 1000 bet, you expect to win 1150. Of course, you will lose 1000 when he has the hands that beat you and you don't draw out, and win his 1000 plus the 500 in the pot when he has something that you beat. But over the long term, presented with this bet over and over and over, he will have different hands each time. Sometimes, you're crushed; this time, maybe, you're crushed. But you can't know that. So you push, he calls, and...

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I just need to add a sentence. When we say that 46% of the time you will win the pot, we can also say that your "equity" in the outcome is 46%. Your equity in a hand will shift from street to street, but you always have equity unless you're drawing dead. We say that you have equity vs a particular hand or equity vs a range. Think of equity as your share of outcomes and you have mastered it. Here's a quick example that should make it clear. You have 1000 chips. Villain has 1000 chips. You are going to get it in and he will call. So there are 2000 chips at stake. Now, you figure you will win 60% of the time. So your expectation is .6 x 2000 = 1200 chips, or +200. So we say your EV is 200. It's worth 200 to you to get it in with this guy. Remember, you will not win 200 ever. Only 1000 sometimes. Your equity can be described as 60%, because your "share" of the outcome is 60%, or 1200. But note that equity and EV are not the same thing. Say you and villain were heads up before this hand and it's a winner takes all tourney. Your equity in the tournament was, of course, 50% before you pushed, and will be 100% or 0% after.

So to be clear, equity is your share in the outcome of the tourney or hand, EV is the change in your equity that an action will bring.