Poker Book Strategy: Pot Odds and Hand Analysis – Harrington on Hold’em Volume 1

Harrington on Hold’em is a series of poker books about poker strategy, particularly for Texas hold’em poker tournaments. They were all written by Dan Harrington and Bill Robertie and published by Two Plus Two Publishing. Volume 1: Strategic Play contains seven key sections and focuses on the basics of poker, such as playing styles, starting hands, pot odds, and hand analysis. The book provides poker strategies for every phase of tournament play, covering the early phase where the stakes are small to later tactics such as bluffing, flops, scare cards, playing shorthanded, loose games, and endgame play.

Here we are taking up the fourth part of the book – Pot Odds and Hand Analysis which is spread across 24 pages dedicated to the theoretical strategy, 4 examples and 10 hands scenario problems. We run though only a few examples given in the book, for complete learning, we suggest going through all the examples and problems given in this book.

Let’s read Harrington in this chapter below:

All successful gambling is based on one simple idea: making good bets at favorable odds. Assessing whether a bet is good or not involves knowing two key facts:

1. What are the odds against your winning the bet?
2. What are the payoff odds if you win?

When the payoff odds are higher than the odds against your winning, you have a good bet. Over time, if you kept making the same bet, you would win money, although the fluctuations might be severe. When the payoff odds are lower than the odds against your winning, you have a bad bet. In time, if you keep making such bets, you will lose money.

As a simple example, suppose that someone offered to wager on the roll of a single fair die. You’re willing to bet $1 that you can roll a six. He bets $6 that you can’t. You eagerly take the bet, because it’s slightly favorable for you. There are six possible outcomes, one which wins for you and five which lose. The odds against your winning are 5-to-1. But your payoff odds are 6-to-l-Since the payoff odds are higher, the bet is profitable for you. (In six average trials, you will lose $1 five times and win $6 once, for a net profit of $1, or almost 17 cents per trial.)

When you walk into a casino, you are confronted with an avalanche of possible bets, almost all of which are unfavorable. Although most people who frequent casinos are aware that they are betting against the odds, only a few really understand what that means. Casual players mostly imagine that they lose when they place a bet and it doesn’t win, or that the games don’t let them win often enough. Actually, that’s not the case. If you bet on the number 22 on a roulette wheel, the number will hit, in the long run, exactly as often as it’s supposed to (one time in 3 8), and when the number doesn’t hit, the casino is perfectly fair – it takes all your money, just as it’s supposed to. In fact, in a sense you lose only when you win. When number 22 actually hits, the casino pays you less than required for an even-money bet – 35-to-1 instead of the even-money odds of 37-to-1. It’s these tiny taxes on the winning bets that provide the casino with all its gambling profits. When you walk through a casino into the poker room and sit down at a no-limit hold’em tournament, the picture changes a bit. As the tournament goes on, you’ll be confronted with a long string of possible bets. The good news is that some will be favorable and some will be unfavorable, but you get to skip the unfavorable ones and concentrate on the favorable ones. The bad news is that it won’t be obvious at first which is which. Figuring that out is up to you, but the information in this chapter will put you quite a bit ahead of most of the other players. Becoming a better player is really a matter of recognizing and making your favorable bets, while avoiding the unfavorable or break-even bets.

Analyzing a Poker Bet: The Two Parts

To figure out if a bet is a good one or not, you need to know the payoff when you win, and the odds against your winning. In poker, the payoff when you win is revealed by the pot odds. How much is in the pot, and how much does it cost you to play? The odds against your winning comes from an intelligent analysis of what’s happened in the hand so far.

Neither part is easy, but both are doable and can be learned. Of the two, calculating or estimating the pot odds is simpler, so we’ll start with that.

Pot Odds

If your opponent has put you all-in, or has made a bet which is the last significant bet of the hand, then the pot odds are easy to calculate. Just calculate or make your best estimate of what’s in the pot, and compare it to the amount required to call. The result is your pot odds to call.

Example No. 1. The pot currently contains $900. You have $600 left, and are in the hand with one other player. The other player moves his last $1,000 to the center, putting you all-in. What are your pot odds?

Answer: Since you only had $600 left, your opponent can only actually put $600 in the pot. The pot you’re shooting at is therefore $1,500, and it costs you your last $600 to call. Your pot odds are therefore $1,500-to-600, or 2.5-to-1.

Example No. 2. It’s fourth street, and the pot contains $1,000 chips. You hold the Q♣ 8♣

and the board is A♣ K♣ 7♦ 6♥

Your opponent, whom you believe to have a high pair, bets $500. You both have more than $3,000 in your stacks. What are the pot odds?

Answer: This hand looks similar to the last one, but contains a few more ideas. We can quickly see that after our opponent’s bet, the pot will contain $1,500, and it will cost us $500 to call, so the pot odds are l,500-to-500, or 3-to-l. But unlike the last example, this may not be the end of the betting for the hand. There could be another bet on fifth street. How does this effect our calculation?

Expressed Odds and Implied Odds

The odds of 3-to-l we calculated in the last example are the expressed odds, the odds that are currently being presented by the pot. Just as important, however, are the implied odds, the odds that will eventually be offered by the pot after all the betting is done.

In many hands, the expressed odds will not justify a call, but the implied odds will. In our previous example, for instance, you are drawing at the nut club flush. If you hit that flush, and you have estimated your opponent’s hand correctly, you will win. If you miss, you will lose. But if you miss, you won’t have to invest any more money in the pot, whereas, if you hit, you may win some more money from your opponent. That’s not certain, since the third flush card will be visible on the board, and your opponent may not want to call a large bet. Suppose you believe that he will fold a large bet on the end if the flush card hits, but he will probably call a smaller bet, say one of $500. In that case your implied odds for your call on fourth street would be the $1,500 currently in the pot, plus the extra $500 you could win on fifth street, measured against the $500 required to call, or $2,000-to-$500, which is just 4-to-l. That’s about the odds of making your hand, so the call is reasonable (but wrong because he will not always call the bet). To be more precise in your calculation multiply the size of your intended bet by the probability he will call it. The resulting “expected value” of his call should then be added to the pot to get your implied odds. Obviously you should also use this method to determine how much to bet in the first place. Normally you should choose the bet with the higher expected value, though in a tournament you might choose a slightly smaller expected value to get a surer call.

In no-limit hold’em it is often correct to accept slightly unfavorable expressed odds to draw at monster hands that can win all your opponent’s chips on the river. This is even more true of potential straights than of potential flushes, since straights are easier to conceal.

Hand Analysis

Calculating pot odds is a simple enough business. Look at the pot look at the chips required to call, and make some rough adjustments if there could be more action in the hand. Divide one by the other, and you have your pot odds.

If pot odds is mostly a science, hand analysis is mostly an art. Here you have to figure out what hands your opponent might be playing that would account for his bets so far, and then how likely each of those hands is, and then figure out how likely you are to beat each of those hands, given the hand you have. At the end of this process, you’ll have a probability that you can win the hand. It will necessarily be a rough estimate, based in part on what you know about this opponent in this kind of situation. But it will give you a number to compare to the pot odds you already know, and in most cases, comparing those two numbers will yield a clear decision from a murky situation.

Before we can proceed with some real-life examples, we need a couple of technical details. Let’s take a look at some probabilities of winning certain hand match-ups and the technique Of calculating outs.

Some Standard Pre-Flop Winning Probabilities

In no-limit hold ’em, certain pre-flop winning probabilities arise so often that you need to commit them to memory. If your opponent puts you all-in before the flop, you need to know the likelihood that you can win with your current holding against the various card combinations he may hold. Here are the most common, along with their probabilities and the odds of winning or losing when the hand is played to the end.

1. Higher pair versus lower pair. The higher pair is about 82 percent to win, or about a 4.5-to-l favorite. The most favorable situation for the higher pair is to be close in value to the lower pair. For instance, a pair of kings is a slightly bigger favorite against a pair of queens than against a pair of nines, because the kings interfere with some of the straights that would help the queens pull ahead. The presence of common suits also helps the higher pair slightly. In pair versus pair with two suits in common, the higher pair gains about 1.5 percent in comparison to pair versus pair with no suits in common.
2. Pair versus two higher cards. This is the basic “race” situation that you see so often in all-in showdowns in no-limit hold ’em. The pair is about a 55-to-45 favorite. The pair does slightly better if it’s close in value to the high cards, by eliminating some straight possibilities. The presence of common suits also helps the pair. (Obviously the unpaired hand is helped slightly if it is suited or connected.)
3. Pair versus two lower cards. Interestingly, if the two lower cards are close together in value this is only slightly more favorable than high pair versus low pair. The increased chance to make two pair and the increased chance to make a straight largely compensate for the loss of trip possibilities. The higher pair is about a 5-to-l favorite.
4. Pair versus a higher and lower card. The pair is about a 5-to-2 favorite. The chance of a straight drops since the pair is taking away two key cards, but there’s about a 30 percent chance of just pairing the overcard which is mostly good enough to win.
5. Two higher cards versus two lower cards. The two higher cards are about a 5-to-3 favorite. This statistic always surprises beginners, who when they see ace-king against something like eight-six imagine that the ace-king must be a huge favorite. But whoever makes a pair mostly wins, and the edge for the ace-king just comes when no one makes a pair or both make a pair.

Calculating Outs

If after the flop you believe that your hand needs to improve to win, you’ll have to calculate your outs. An “out” is simply a card which, if it hits on fourth street or fifth street, will improve your hand to a winning hand. Calculating your outs will give you some idea of how big an underdog you are at the moment.

When you calculate outs, always keep in mind that it’s an imperfect business at best. If your opponent is bluffing, your hand, weak as it is, may already be best. If your opponent is trapping with a full house or four of a kind, what you believe to be outs may only be cards that cost you your entire stack. But with those caveats aside, it’s still very useful to have some idea of how many cards could give you a likely winning hand.

Example No. 1. You hold A♥ K♥

and the flop is J♥ 9♠ 5♥

Your opponent called before the flop but now bets strongly, and you suspect he has at least a pair. How many outs do you have?

Answer: Since there is no pair on board, your opponent cannot have a flush or a full house yet. So if you make a heart flush, you can be reasonably sure of winning. (You hold the ace, so even if your opponent is also drawing at a heart flush, you have the nut flush.) There are four hearts visible to you and nine others remaining in the deck. We’ll count those nine hearts as nine full outs.

If your opponent has only a lower pair, then an ace or a king will also qualify as an out. But if he now has two pair or a set, then hitting an ace or a king won’t help you. His most likely hand here is a pair of jacks, but he could have called your bet with jack-nine, a pair of nines, or a pair of fives. You can’t count the three remaining aces and three remaining kings as six full outs; probably counting them as an average of four outs is most reasonable.

There are sequences of two running cards that could also help you. A queen and a ten on the next two cards gives you the highest possible straight. The chances of that are pretty small, however. Although there are four queens and four tens, two of them were counted already – the Q♥ and the T♥. So you would have to hit one of the six remaining cards on fourth street, followed by one of five cards on the river. That amounts at best to a fraction of an out.

So our best estimate of your total outs is about 13. With two cards to come, that makes you a slight underdog in the hand.

Certain common situations arise so frequently that you should become familiar with them. Here’s a small table of these situations and their associated number of outs.

A good rule of thumb to remember is that if you have 14 outs, and you know you can see both the turn card and the river card, then you’re about even money to win the hand (as long as you will almost always win when you hit). The only time you can know that, however, is when you’re facing an all-in bet after the flop. Now a call will let you see the hand down to the end, so 14 outs will indeed make you about even money. But if you’re facing a bet after the flop and both you and your opponent have plenty of money left, that assumption doesn’t hold. In that case, missing your draw on fourth street may mean that you’ll be facing a large bet which, with only one card left to come, you won’t be able to call.

With those technical parts behind us, we’re now ready to look at some real hands and see how all these ideas fit together in practice.

Example No. 3: You’re playing an online tournament and just five players remain. The blinds are now $50/$100, and you have $4,300, a bit less than half the chips at the table. On the button, you pick up A♣ 8♦.

The two players in front of you both fold. You raise to $200. The small blind folds but the big blind, a solid player with $1,100, calls, putting in another $100. The pot is now $450. The flop is 5♠ 3♠ 2♦.

The big blind checks, and you bet $250, a little over half the pot. The big blind raises you to $500. What do you do?

Answer: Your first thought should be to throw your hand away. You did your bit, but a check-raise indicates strength, and all you have right now is an асе-high hand.

But before you fold, notice that it costs you only $250 chips to play a $1,200 pot. With almost 5-to-1 odds staring at you, it’s worth spending a little time calculating your outs. Let’s see what they are.

1. An ace might be an out, and there are three aces left in the deck. But it’s not an out if your opponent is holding a higher ace-x.
2. An eight might be an out, and there are three eights left in the deck. But it’s not an out if your opponent was slowplaying a higher pair.
3. A four might be an out, and there are four fours left out there. But if your opponent has an ace like you do, then a four gets you only a split of the pot.

So how many outs do you actually have? If your opponent check-raised with something like 7♥ 7♣

then you have ten outs. But that’s a somewhat weak holding for a check-raise, and the truth is that you have no idea how many outs you have. Remember too your opponent may have actually made a strong hand, in which case you may have zero outs.

This looks like a fold to me.