by Lou Krieger

First published in Poker Player Newspaper

Assuming you don’t hold any of the cards in question in your hand and there is no raise before the flop, what do you regard as more dangerous in a hold’em game: an ace on the flop, or a flop that contains either a ten or a jack? That was the subject of a discussion I had recently with a well-known poker theorist. One line of reasoning we discussed is that an ace is more dangerous because so many players are prone to play aces – any ace at all – and it doesn’t much matter what the side card is.

Even if you’ve never thought much about this before, the next time you sit in a hold’em game take a look at the hands being turned over in a showdown. While you’ll frequently find players holding A-6, A-2 and hands of that ilk, you seldom see anyone turning over K-2, Q-3, J-4 and similar hands unless they got a free play in the big blind.

While poker players are more prone to see a flop with an ace in their hand than a hand containing a jack or a ten, there’s another school of thought that says a ten or a jack is more likely to be in an opponent’s hand when the pot has not been raised, because those kind of cards – cards that lie near the bottom end of the playing zone – are played more often for one bet, particularly by players at the rear of the betting order.

Another line of reasoning suggests that the time you have to worry about your opponent holding an ace in his or her hand is when the pot is raised before the flop. With aces, those who follow this line of reasoning are either going to fold because their kicker is weak and they are out of position, or they are going to raise because their position is good, their kicker is strong, or they might be able to win the pot right then and there just by raising.

My personal opinion, and I have no hard, quantifiable evidence to back this up – it’s simply anecdotal data gathered over a lot of poker tables – is that it really depends on the nature of the game.

In lower limit, no-fold’em hold’em games, where you frequently see four or more players taking the flop, some of those players will be in there with any ace. Period. End of story. A-2? “No problem, man; I’m calling!”

Ace-anything, it seems, is worth a look at the flop by many players, particularly if the pot was not raised. In fact, even when it is raised there are lots of players who will call in early position with any ace. They often wind up cold-calling a raise only to see a flop with a weak ace regardless of the position they are in.

In bigger games, and for want of something more precise, let’s say we’re talking about games of $15-$30 and above, players usually bring their ace in for a raise or they fold. If the pot was not raised before the flop in a higher limit game, and an ace flops, you can discount it much of the time.

But that’s not the case if a ten or a jack appears on the flop, because in bigger limit games those are precisely the kind of hands many players will limp in with as long as they have a reasonable assurance of seeing the flop for one bet – and this reasonable assurance translates into playing a ten or a jack or a jack and a ten from late position with a large field for only one bet.

In a mid- or big-limit game, if the pot is not raised and an ace flops, I usually discount it for any hand other than one the blinds who were lucky enough to see a free or inexpensive flop. That makes a flop with a ten or jack scarier in an unraised pot. But in lower limit games, an ace on the flop stands a good chance of matching one in someone’s hand.

by Lou Krieger

First published in Poker Player Newspaper

The flop is hold’em’s defining moment. For the cost of one small bet, you get to see 71 percent of your cards on the flop. That’s right; the flop comprises five-sevenths of your entire hand. It’s one of poker’s biggest bargains. Because of that, it’s imperative that the flop fit your hand in order to warrant the relatively pricy decisions to see the turn and the river. Deciding to see the turn or the river means seeing fewer cards at a higher price, and you need to have something that makes your investment worthwhile before making this decision.

Fit or fold – a phrase coined by poker author Shane Smith – means a couple of things. The first example is obvious: The flop should help your hand. If you begin with Q-J and the flop is Q-J-3 you’ve made two pair and were obviously helped by the flop. In fact, you can say it fit your hand like a glove. If you began with a weak hand such as 4-4 in the big blind and flopped a set of fours, the flop smiled on you in an even bigger way.

But that’s not the only way the flop can fit your hand. If you started with 9-8 and the flop was T-7-3 you flopped four to a straight and will probably get the right price from the pot to justify playing on in hopes of making a straight. The same holds true if you began with two suited cards and were fortunate enough to find two more cards of your suit on the flop. Now you have a four flush and that’s usually a draw worth playing too.

In addition to a flop that fits by pairing your hand or providing a draw to a big hand, the flop can help you by not assisting your opponents at all. Suppose you raise with a pair of queens before the flop. You’re rooting for a third queen on the flop, but there are a lot of cards you’re rooting against too.

Even though you were the raiser before the flop, you’d like to dodge a king or an ace, because those cards can give an opponent a pair higher than your queens. You’d also like to duck two or three mid-range or big adjacent cards because they increase the possibility of someone making two pair, a straight draw, or what’s worse, a straight. If the cards are two-suited you have to fear a flush draw, and if they are all one suit another player might already have a flush.

But if the flop is J-7-3 of mixed suites it effectively fits your hand because it probably missed your opponents’ hands. Unless someone has flopped a set – and the odds are against that -your hand, which was almost surely the best one before the flop, is probably still in the lead with only two cards to come.

You’re in good shape at this point. You can bet and force any opponent with a lesser hand to take the worst of it if he or she decides to call. With the best hand, betting gets more money into the pot, and that’s a good thing too. You are building a pot you are favored to win, while making it more costly for any opponent to stick around in hopes of outdrawing you. When you’re playing Texas hold’em, here are three rules of thumb for playing the flop:

• Play if the flop improves your hand right now.

• Play if the flop provides a draw to a straight or a flush that figures to win the pot if you complete it.

• Play if you have the best hand before the flop and the flop is so ragged in texture that it figures to miss your opponents’ hands as well as your own.

If none of these conditions are present, you can consider the flop to have missed your hand, and you have no reason to be in the pot unless you have a valid reason to believe your opponent is likely to fold to a bluff. But that’s a different story for a different day.

by Richard Burke

First published in Poker Player

On a blustery Friday afternoon in late winter, I was playing $4-8 Hold'Em in my local poker room and wondering if the dealer would ever toss a starting hand my way, when Steve came over to my Hold'Em table between hands at his table. He asked me about a hand in a ten-handed $2- 5 No-Limit Hold'Em game the night before.

The Flop was Af-Qa-9d. Holding pocket Aces, Queens, and Nines, three players went all-in before the Turn. They tabled their Sets face-up, crossed their fingers, and watched blanks fall on the Turn and River. (The dealer pushed the $2500 pot to the player with the Aces.) Steve asked me, "What were the odds of three players making Sets on the Flop?" If you're not interested in combinatorial mathematics, then skip on down to the answer. For those who are, here's how I figured it.

Consider first that three pairs of three different ranks were dealt among the ten players. There are 286 ways to choose those three ranks. Counting suits, there are 6 ways to deal each pair. There are forty cards of other ranks available to distribute among the seven other players, and C(40,14) ways to do that. There are 13!! ways that seven doubletons could have been dealt to the other seven players. The total number of ways to deal three pairs of different ranks and fourteen cards from the other ten ranks is 286*63*C(40,14)*13!!, or 193,733,765,427,641 million. The formula, C(52,20)*19!!, gives the total number of ways to deal out ten doubletons from a fifty-two card deck, or 82,492,346,176,096,200 million. (The large numbers are unwieldy; I'll use probabilities hereafter.) Dividing obtains the probability, 0.0023485. It remains to compute the probability that one of the two remaining cards of each rank would appear on the Flop. The chance of that is 23/C(32,3), or 0.001613. Their product is 0.0000037879.

If seven cards of three ranks were dealt among the ten players, then there are 858 ways to choose the three ranks, and 4*36 ways to choose three pairs from those ranks. 858*4*62*C(40,13)/ C(52,20)*3*13!!/19!! is the whole formula, which obtains 0.007306. The Flop must contain the one rank, and either of the two cards from the other two ranks. C(1,1)*C(2,1)*C(2,1)/C(32,3) obtains 0.008065. The combined probability is .000058923.

For eight cards having been dealt from three ranks, 858*42*6*C(40,12)/C(52,20)*9*13!!/19!! obtains the probability that they will have been dealt into three pairs, 0.006785. The Flop must contain one from two ranks and either from the other rank, a probability of C(2,1)/C(32,3), which equals 0.0004032. The product is 0.0000027357.

If nine cards were dealt, there must be three from each rank, 286*C(3,3)*C(3,3)*C(3,3)*C(40,11)/ C(49,20), and the probability that they were dealt into three pairs is (27*13!!/19!!), for a total probability of 0.001872. One of each rank on the Flop has a probability of 0.0002016. The combined probability is 0.0000003773.

The inverse of the sum of the probabilities is 78,166. The odds against three players flopping a Set are 78,165 to 1.

by Matt Hilger

First published in Card Player

There are always two key decisions you must make when taking the initiative in no-limit hold’em while playing a hand. Are you going to play, and if so, how much do you want to bet? The amount of your bet is always very important. If you bet too little with a good hand, you could be inviting your opponent to hit a long shot that might cost you your entire stack. If you bet too much, you might find yourself in a situation in which you are winning only small pots and losing large pots. A key part of the decision in determining how much to bet depends on the pot odds and implied pot odds.

This column looks at how to use odds and probabilities when deciding how much to bet. Realize that other factors such as psychological warfare are important in no-limit. Sometimes we might intentionally bet a big hand small, hoping that our opponents will sense a bluff and try to make a play at us. At other times, we might bet big, trying to indicate a bluff. Both are ways of using psychology during a hand. Putting the psychological part of the game aside, the first step is learning the mathematical foundation of how much you ought to bet. Once you master the foundation, you can then learn more advanced plays in terms of the psychology to use against each individual opponent.

Protecting Your Hand
The amount you bet on the flop is a critical decision in no-limit hold’em. Let’s start with an extreme example just to demonstrate a point. You are heads up in a major poker championship and are dealt J-J. The blinds are $10,000-$20,000 and you raise to $75,000 preflop. A very tricky, loose opponent calls. There is now $150,000 in the pot. The flop comes J-7-3 rainbow. You both have about $880,000 remaining in chips. How much should you bet?

Of course, there are always lots of factors that go into betting in no-limit, but we are going to look at one of the most critical factors, which is protecting your hand. You have flopped top set and the board is relatively harmless. Assume that you decide to slow-play and bet $40,000. Your opponent calls, and the turn card is an 8. Then, in a flurry, you are both all in. You turn over your set, only to look in exasperation as your opponent turns over 10-9 for a straight.

Your opponent was getting almost 5-to-1 pot odds to call on the flop; however, implied pot odds are critical in no-limit, as you can sometimes break your opponent, or vice versa. In this case, your opponent is getting good implied pot odds when he has to call a bet of only $40,000 in hope of hitting his miracle 8. He was 10.5-to-1 against hitting the straight. He must think he can win $420,000 to justify calling. There is already $190,000 in the pot, so he must be able to extract $230,000 more from your stack to justify calling. In this case, if you have a big overpair, there is a good chance that he can take your whole stack if he hits a huge hand. By betting too little, you put your opponent in a position in which it is correct to call. Protecting your hand is crucial to your survival in no-limit hold’em tournaments.

How should you protect a hand? Let’s go back to the example in which you have $880,000 in chips remaining. First, evaluate the flop to identify any dangers. Note that all of the following hands have gutshot possibilities: 10-9, 10-8, 9-8, 6-5, 6-4, and 5-4. This is actually a decent number of hands that you need to worry about. A gutshot is approximately 10.5-to-1 to improve; therefore, if you want to protect your chip stack, you must bet an amount to protect the amount in your stack and what is already in the pot. With $880,000 in chips and $150,000 in the pot, you can divide the total, $1,030,000 by 10.5, which equals roughly $98,000.

If you bet slightly more than this, something like $110,000, you ensure that your opponent is not getting the correct odds to call. If your opponent calls, he is making a mistake. There is no guarantee that you will win, but at least you are forcing your opponent into making a mistake because of the amount you bet. Of course, you could also go all in, which would really discourage a call, but you also want to encourage action in order to give yourself a better chance to win a big pot. This balance of risk and reward is a fine line that is important to evaluate when making decisions.

If your opponent hits a pair on the flop, such as a hand like 8-7 with the J-7-3 flop, he might call, thinking he has the correct implied pot odds to draw to two pair. If he hits two pair or trips, you will break him. When you protect your hand, you actually want your opponent to call, since he is making a mistake in doing so. Note that even if you bet slightly less than $98,000, an opponent should not call, as he cannot be sure that he can extract your entire stack. The $98,000 is simply the mathematical amount that truly protects your hand.

Try to bet amounts that will put your opponents into situations in which they will make mistakes by calling.

There are three steps to take when determining how much to bet to protect your hand:

1. Calculate the pot odds and the implied pot odds based on the smallest stack between you and your opponent.

2. Determine potential dangers with the flop versus your hand.

3. Bet enough that your opponent’s effective implied pot odds do not justify calling, given his odds of improving to the winning hand.

The second step deserves more discussion. A key part of it is understanding the texture of the board. My first book, Internet Texas Hold’em: Winning Strategies From an Internet Pro, looked at various types of flops in detail, and the potential dangers of each. Are flush draws possible? Could someone have an open-end straight draw? What about a gutshot-straight draw? Could our opponent have five outs to two pair or trips? Maybe it is likely that our opponent has only three outs in a scenario in which we both hold top pair (A-K vs. K-Q with a flop of K-8-2). All flops and all hands are not created equal!

Understand that protecting your hand is not always the only consideration. Sometimes you may be unsure that you have the best hand. For example, you might have an overpair and your opponent could have a set. In these cases, you have to balance how much you want to protect your hand with the possible risk that your opponent already has a better hand.

Also realize that protecting against implied pot odds does not always mean protecting your entire stack. You should bet an amount to protect against only the total amount of chips you are willing to commit to the pot. For example, sometimes your opponent might call a bet and hit a draw, and you would fold. In these cases, you are not giving your opponent implied pot odds, since you are not committing more money to the pot.

Odds and probabilities have many applications at the poker table. Psychology is still important, but be sure to use the odds as a foundation to help you make better decisions when determining how much to bet.

Matthew Hilger is the author of Internet Texas Hold’em. This column contains excerpts from his upcoming book, Texas Hold’em Odds and Probabilities: Limit, No-Limit, and Tournament Strategies. You can ask Matthew questions in his poker Forum at www.InternetTexasHoldem.com.

The variables that must be considered

by Matt Matros

First published in Card Player

Poker has become so mainstream that it’s actually possible to place wagers through online betting establishments on who will win most major poker tournaments. I wonder if the day will come when we’ll walk into Vegas sportsbooks and see “World Series of Poker Odds” staring back at us in those big red lights. Since this is supposed to be a poker math column, I thought I’d take a look at how to interpret the published odds for poker tournaments, and how to handicap tournaments ourselves.

What are the chances that a given player will win a poker tournament? Well, for an average player, his chances of winning a tournament are 1/N, where N is the number of players in the event. So, in a 10-player tournament, the average person would have a 1-in-10, or 10 percent, chance of winning. But wait, what is an average player? Let’s say in our 10-player tournament that there is one exceptionally good player who is twice as good as the average player in the tournament. That means he has a 20 percent chance of winning. Let’s say furthermore that the other nine players are all about equal. That means each of those nine players has an (80/9) percent = 8.9 percent chance of winning the tournament. So, only one player in the entire field is an above-average player! In practice, this means that only the very good player should expect to make money in this 10-player tournament. The other nine players all have a negative EV (expected value).

What if we vary it a little bit and say that in addition to the one very strong player in the tournament, there is also one truly terrible player. The truly terrible player has half the chance of winning as an average player; he’s only a 1-in-20 shot (5 percent) to win the tournament. That still means that each of the other eight players, who are all about equal, has only a 9.4 percent chance of winning. It would take two truly terrible players in the field (with 5 percent chances of winning each) to offset the effect of the very good player. With one very good player, two terrible players, and seven equally skilled players, the seven middling players would once again have a 1-in-10 chance to win the tournament, making them average players, and zero EV (assuming no rake — ha!).

The point of all of this is that, for the average player, it usually takes several terrible players to make up for just one excellent player in a given tournament. We all know that terrible players find their way into poker tournaments all the time, but are there really enough of them to overcome the excellent players?

Let’s try to make some estimates to answer this question. If there are 6,000 players in the 2006 WSOP championship event, and you could perfectly rank them in skill level from 1 to 6,000, what player would you have to get down to before your group of players had a 20 percent chance of winning the tournament? 300? 600? 1,200? If the answer is 300, that means the average player among the top 300 would win the WSOP four times as often as he would if the results were based entirely on luck; 600 means the talent isn’t quite so top-heavy, and that the top 600 players win only twice as often as they’d win by chance; 1,200 means the tournament is all luck, and the top 20 percent of the field is no different from the bottom 20 percent of the field. So, how much skill is there in the WSOP main event, and how good are the very best players? No one has a definitive answer to this question. My personal opinion is that we’d need to pick the top 500 players before we would have, collectively, a 20 percent chance to win the tournament. In other words, I think most very good players win the tournament only about two and a half times as often as the average player does. Bear that in mind when making your wagers.

Next question: Starting from the bottom of the ranking list, how many players would we need to pick before we’d have a 10 percent chance to win the tournament? This question is perhaps even trickier than the first one. There are some people who play so badly (perhaps they’re playing the WSOP for fun, having never played poker before) that they essentially have zero chance of winning the tournament. And there are some players who have a clue about hand values, but have such large flaws in their games that they are at an enormous disadvantage. And then there are those who, while clearly below average, play a style that at least gives them a chance to win if they get run over by the deck. My personal estimate is that the bottom 1,500 players would collectively have a 10 percent chance to win the event.

These estimates leave 4,000 players in the middle, and to keep things simple, let’s say they’re all relatively equal in skill level. That group of players collectively has a 70 percent chance of winning the tournament. Individually, a member of that group can expect to win the tournament once every 5,714 tries. That’s better than average! So, in my opinion, there are enough bad players in the WSOP main event to offset the effect of the very good players, making it a profitable investment for the merely good players. (I hope this is encouraging to a lot of readers.)

Now, how does this affect gambling on poker? Well, let’s say we find a sportsbook that offers lines on 500 players for this year’s WSOP (I don’t think this is such a bad assumption, by the way). According to my above estimates, the average player on that list should have a line of about 2,500-1, if (and this is an enormous if) the sportsbook has accurately listed the 500 players most likely to win the tournament. In real life, there is no way anyone could know who the best 500 players in the WSOP are. Bearing this in mind, I find it hard to imagine that there should be more than 50 players deserving of a 2,500-1 line. I don’t think more than a handful of players should be listed at 1,500-1. And there isn’t a player in the world whom I’d consider a good bet at less than 1,500-1 (well, maybe Phil Ivey).

Betting the “field,” or betting that one of the unnamed players will win the tournament, is an interesting proposition. Assuming that the 500 named players are more likely than 500 random players to win the tournament, but less likely than the top 500 players would be, I’d guess that the 500 named players would collectively have about a 15 percent chance of winning the event. That leaves the other 85 percent for the field — making the appropriate odds for betting on the field 1-5.7. I’d be surprised if any sportsbook offers better than 1-9 (meaning you’d risk $900 to win $100) on the field in the 2006 WSOP. But if you do find a field wager in the neighborhood of 1-3, in my opinion you should scoop it up, assuming the sportsbook has set lines on fewer than 10 percent of the players in the field.

In summary, when considering betting on a poker tournament — and, indeed, when considering whether to enter a poker tournament — be sure to think of (1) the percentage of truly great players in the field, (2) the percentage of clueless players in the field, (3) the number of entrants in the event, (4) the “skill factor” of the event (events with more play generally enable better players to increase their edge over the field), and most obviously (5) the relative skill of the actual players you might wager on. Notice that I spent no time giving you my opinions of the relative strengths of today’s name poker players. Don’t expect that pattern to change in these pages anytime soon.

Matt Matros is the author of The Making of a Poker Player, which is available online at http://www.cardplayer.com/.

by Richard Burke

First published in Poker Player

On a busy Friday afternoon in mid-winter in my local poker room, I folded my hand in a ten-handed $4-8 Hold’Em game. Just on my left, LindaMae raised and everyone except the Big Blind folded. The Flop came [Ah]-[6s]-[7d]. The Big Blind bet $4 and LindaMae mucked her pocket Kings. “Damn Ace-magnets!” she swore quietly. While I waited for the next deal, I wondered whether Kings really do attract Aces the way funerals draw politicians. Given that she held pocket Kings, the probability that one or more Aces would flop is (1-C(4,0)*C(46,3)/ C(50,3)), or .2255. So LindaMae was unlucky on that Flop, because three times out of four no Ace would fall: 77.4% of the time the Flop would be Ace-less. Furthermore, if there were no Ace on the Flop, then the probability of no Aces on the Turn or River is given by C(4,0)*C(43,2)/C(47,2)), or .8353. So, if an Ace hadn’t flopped, then five times out of six there wouldn’t be one on the Turn or River either. Putting those two events together, starting with pocket Kings, the chance is 64.7% that there won’t be an Ace on the table after all the cards are out. Pocket Kings are NOT Ace-magnets; it just seems that way.

(With no Ace on the table, it’s a little more likely that one or more of her nine opponents would have been dealt pocket Aces. That chance is .0543, about 1 in 19.)

The table shows the probability that there would be exactly 0 through 4 Aces on the tableau. The table also shows the probability that 0 through 4 Aces would be dealt among the players versus the number on the tableau.

The table shows that for exactly one Ace on the tableau, it’s a 21% chance that no Aces were dealt among the nine opponents: so, it’s a 79% chance that one or more Aces were dealt. Because most play Ace- Any in low-limit Hold’Em games, when there’s exactly one Ace on the tableau, four times out of five somebody will have a Pair of Aces or better.

I wouldn’t have criticized her for calling one small bet, because her pot odds were larger than her cards odds: counting the house rake, she had pot odds of $24 for $4, better than the about 1 in 5 odds that an Ace wasn’t dealt. If the Turn card didn’t improve her hand, then counting the rake and bad-beat drop, her pot odds to see the River card would have been $40 for $8, about equal to 1 in 5. A call there wouldn’t have been all that bad either.

LindaMae reasoned differently. The Big Blind was a solid player and because of her early-position, pre- Flop raise, he surely put her on a big Ace or big pockets. She reckoned that: a) he wouldn’t have bet into her post-Flop unless he had a good hand, e.g., Two Pairs or a Set; or b) he had a weak Ace and was probing to learn if she had a better Ace. Either way, headsup and having only two outs, her Kings were headed for the muck. I couldn’t disagree.