Basic Poker Math, copied from a web page. This is good reading and very interesting material.
I know that a lot of you are less than thrilled by mathematical dissertations, but it's all a part of the game and you must have a grasp on at least a few basic principles in order to be successful at Hold 'em Poker, so please bear with me. I'll try to make this as simple, easy to understand and brief as possible. If you're a student of my Blackjack School, you're hopefully already familiar with the term, "expected value" (EV), but it's not something you hear about a lot in the poker world. For whatever reason, most poker players, authors, commentators and so forth seem to prefer using "odds" to describe a situation. For example, a particular play may have odds of "4 to 1 against", which basically means it has a 20% probability of happening.
The terminology of odds have always confused me and because of that, I wanted to teach myself, and you, a quick way of doing calculations in your head, so I've decided to go more with probability when calculating EV, rather than odds. I mean, does 5 to 1 odds mean a 16.67% probability or a 20% probability?
While there's not a huge difference between the two, being consistently wrong about how you figure your chances in a given situation will eventually cost you some hard-earned $$$. But for those of you who'd rather deal with odds, let me show you the easiest way to convert probability to odds. Any probability that's expressed as a percentage can be converted to odds by first subtracting the probability from 100, then dividing the result by the probability.
For example, in the case of a 16.66% probability it'll look like this: 100 minus 16.66 = 83.34 divided by 16.66 = 5.00 or 5 to 1 "against". In the case of a 20% probability, it'll look like this: 100 minus 20 = 80 divided by 20 = 4.00 or 4 to 1 odds "against".
What do 5 to 1 odds "against" mean in the real world? Well, it means that for every 6 times you try the whatever you're talking about, it'll work once. More confusion, right? The clue for getting a good grasp on this is to add the 5 to the 1 to get 6. Out of 6 attempts, 1 will work, so the odds are 5 to 1 "against." Isn't it really just more simple to say you have a 16.66% chance of success? That's what I'm going to do as I take you through this course, use probability in conjunction with bet size to arrive at EV (expected value, remember?).
For example, if your $10 bet has a probability of success of 20%, your EV is $10 x 20% (or 0.20) = $2.00. It's what we do in Blackjack all the time; a hand of 6,4 versus the dealer's 7 has an EV of -.476 if you stand (!!!), an EV of +.293 if you hit and an EV of +.406 if you double. It's just a matter of choosing the highest EV in the play of your hand, so you should double 6,4 vs. 7.
Unfortunately, it doesn't work exactly that way in Hold 'em Poker, because your hand is always being compared to the other players' hands and, as the old saying goes, "Any hand can be a winner in poker". Rather than measuring the value of a given hand, I'm going to show you how to evaluate the expected value of your bets with the idea that if you make all (or almost all) of your bets in situations where you have a "positive" EV, you can't help but make a profit. This doesn't mean you're going to win every hand, just like there's no guarantee you're going to win every time you double 10 vs. a dealer's 7. But, if you do it often enough, in the long run you'll make a profit.
Let me give you a quick example of what I mean. Let's say that you hold a hand of 10, J offsuit in the "pocket" in a $10/$20 limit game and the flop comes 10, J, 6 (I'm ignoring suits here). You now have Two-pair and, if you choose to play this hand through to its conclusion - two more cards - there is a 16.5% chance that you'll catch another 10 or Jack, thus ending up with a Full House. (I'll show you how that 16.5% is calculated in a minute.) Now remember that the math can't tell you if the Full House you have is a guaranteed winner because another player may have a higher Full House or Four-of-a-kind, etc. when all the cards have been dealt. But, the math can tell you if betting on your Two-pair makes sense. Let's say all of the pre-flop betting has resulted in a pot total of $60, the bet after the flop comes to you and the pot is now worth a total of $90. Should you make a bet on this hand?
First of all, you have Two-pair, regardless of what happens and that alone may be enough to eventually win, so it has a value of its own. Unfortunately, I cannot (at this time, anyway) assign an EV to that hand, but I'm working on a calculator that will do that. So, for the moment, let's assume that your Full House has nearly a 100% probability of winning the pot, as most Full Houses do. With a 16.5% probability of making a Full House from your hand, the EV of your bet is 16.5% of $90 = $14.85. If the bet you have to make is $10, then you have a definite positive EV and should make the bet. If the bet you must make is $40, it's not as clear-cut a choice.
That's because players betting after you may or may not add more to the pot's value, plus you'll undoubtedly have to make additional bets after the "turn" and "river" cards are dealt. But all we can really do is play our hand one bet at a time, while taking into consideration what other hands are being formed by the other players; don't forget that the flop, turn and river cards belong to them, too.
As we get further into the lessons, I'll show you how to "read" other players' hands by how they bet or don't bet and that will help you in your decision-making process for situations like this where a hand with a positive EV can be suddenly transformed to one with a negative expectation.
Whether or not you make a $40 bet for the hand shown above is immaterial to this situation. What really matters is that you know the probability of making the hand from the flop, forward and you use that to guide your betting. But, and it's a big "but", if you choose to make the $40 bet, be aware that it's probably a negative EV bet and, if you make them often enough, you'll eventually lose all of your $$$. I say "probably" because at this point I cannot precisely quantify the value of your Two-pair other than to say that the only hand it beats is a Pair, but that's often enough to win a pot in Hold 'em. If we somewhat arbitrarily assign a probability of 20% to the Two-pair winning the pot, then the total EV for that hand is about $33 (20% x $90 = $18 + $14.85), so a $40 bet is a borderline decision at best and a $30 bet seems reasonable. However, a $60 bet would be a real "gamble" and you should know that before you make the call.
Some poker experts like to use "implied odds" when making a decision like this and they want you to figure out how many players will call your bet so the total pot before the next bet comes due can be used to calculate your EV, which they call "pot odds". Well, that sounds good and is certainly valid, if you're able to predict just who is going to bet and how much they'll bet. My problems with that concept are many, not the least of which is that it encourages a certain amount of wishful thinking on your part, plus it's yet another layer of calculation that's being added to what is already a fairly complex equation. Just as in Blackjack, I prefer to err on the side of conservatism when $$$ are involved, so rather than use implied odds, I prefer to use the odds presented to me as the hand progresses.
Let's continue along and play out the Two-pair we have by making a $30 call after the flop. Now comes the "turn" card and it may well give us our Full House. But, if your luck is like mine, it won't so we'll have to face more decisions in betting. (If we made the Full House with the turn card, I'm assuming we'll welcome and call any bet or more likely, raise the pot for the balance of play.) With the turn card out, we now have to re-evaluate if our hoped-for Full House can still win the pot. Don't forget that Four-of-a-kind beats a Full House, as does a Straight Flush, so we have to evaluate the impact of the turn card on other players' hands. It didn't help us, but it might have helped them.
If you remember, we had a hand of 10, J and the flop came 10, J, 6. Because I'm ignoring suits in this example, let's rule out the possibility of a Straight Flush, but even if the flopped 10, J were suited, the best anyone could have is a 4-card Straight Flush (called a S.F. "draw") and the odds are greatly in favor of them making either a Straight or Flush, both of which lose to a Full House, so we can't spend our time worrying about losing to a Straight Flush. I've played thousands of hands and have lost to a Straight Flush only one time. But that little, lonely 6 that came on the flop could be a problem if another one shows up as the turn card. It's not inconceivable that some other player has 6,6 "in the hole" and s/he is going to be thrilled to see it, because those Trip 6s will beat our Two-pair if we don't improve. But we have set our course and must go forward, although not blindly. The nice thing about establishing our EV after the flop is that we are done with all of the calculations for this hand, so we can now concentrate on the betting patterns or "texture" of the game.
If a player who has just been passively checking or calling now comes out with a bet or raise, we must take that into consideration when the bet comes to us, but we don't have to refigure the EV because the 16.5% chance of making a Full House applies to both of the last cards.
But, you may ask, what about the bets we already have in the pot; don't they have a place in our calculations? The short answer is "no". Those $$$ are gone, so to speak and we'll only get them back if we win the hand. Think about it: if we don't call, they're lost anyway, so I don't count our previous bets when calculating EV, only the full value of the pot. But, for whatever reason, let's say the turn card inspires a lot of betting and, because this is the first "expensive" round of betting ($20 in a $10/$20 limit game), we cannot just call any and all bets because we have just one more card to come and only a 10 or Jack will make our Full House, so we have four "outs" for the hand. You'll hear that a lot in the poker world; the number of "outs", so let me take a minute to explain it.
Up to this point in our play, we've seen 6 cards; our two "hole" or "pocket" cards and the four community cards on the "board", three from the flop and the turn card. That leaves 46 cards unseen and we can only assume, at least for mathematical purposes, that the two Jacks and 10s that will help us remain in the deck. That, indeed, may not be the case but we have no way of knowing otherwise unless someone shows us their hand, so it's just like in Blackjack; if we don't see it, we don't count it. Of course, we're not counting the cards here (not yet, anyway), so the math is now very simple. Four cards of 46 help us (2 Tens and 2 Jacks) so we have four "outs", or a 2/23 probability of making our hand at this point.
Does this mean that the pot now has to be 11.5 times the size of our bet in order for us to call a bet? Not really, although that would be an ideal situation but it's not likely to occur. By making our bets based upon EV after the flop, it's just like betting when the count goes plus in Blackjack. We don't win with every hand of 20, but in the long run, betting more when we have the advantage and betting less when the casino has the edge gives us the opportunity for profit. In poker, just substitute "casino" with "other players", because that's who will provide the profit opportunity.
Make sense? For every six times we play Two-pair to completion, five won't become Full Houses, so that's why we're only betting one-sixth of the amount we could be betting if we knew that this would work every time. Bets based upon a positive expected value eventually pay off. Maybe not this hand and maybe not even tomorrow, but it will eventually pay. Plus, you now have an unemotional way to play your hand; if it has a positive EV, play it. If not, throw it away. Yes, there will be times when you fold a hand and it would have ultimately won, but to draw another Blackjack analogy, there are also times when standing with your 12 against a dealers up card of 10 would have produced a winner, but we're here to make "percentage" plays, not "intuition" plays. The numbers simply do not lie, either in BJ or Hold 'em Poker, so once you learn to trust them, you're on your way to playing a winning game.
I'm going to give you the percentages of success for making various hands that you may encounter after the flop (5 cards seen), based upon staying with the hand until the end (7 cards seen), but first I want to show you an easy way to check the validity of your bet in the heat of battle, so to speak. If you have a probability of 16.5% in making your Two-pair into a Full House, that means the pot should be at least six times the value of your bet for it to carry a positive EV. Why six? Multiply 16.5% by 6 and you get 99%. A figure of 100% is the threshold of positive expectation, but for me, 99% is close enough because we have some extra EV built into the play due to the possibility of the Two-pair winning on their own. Knowing this little trick will allow you to quickly calculate the pot odds in the manner I've described above by multiplying the bet times 6 and then comparing that figure to the pot total at the time it's your turn to bet. That's very easy to do in a limit Hold 'em game because of the uniform bet size and not so easy in a pot limit or no-limit game. But for now, we're discussing limit Hold 'em, so I won't confuse the issue.
Let me give you an example of how this works. Let's say the pot is $90 and you must bet $10, minimum. Well, six times $10 is $60 and the pot is "paying" you $90, so make the bet. Were the pot only $40, you'd be facing a negative expectation of $20 if you make the bet. Or, if the bet is raised in front of you and you now have to bet $20, six times $20 is $120 and you're only getting paid $90, so you shouldn't make the bet. Conversely, if the pot is, say, $300, you could bet $40 and still have a positive EV. If nothing else, this method of play removes a lot of anxiety from the game; should I call, bet, fold or raise... oh, what to do?
Okay, as promised, here's a chart of probabilities for various hands you might hold after the flop. Remember, the probability of success is based upon you seeing two more cards; the turn and the river and it does not guarantee you'll ultimately win the hand.
|Hand at the Flop||Becomes||At this rate of probability||Bet Multiplier|
|4-card open-ended Straight||Straight||31.5%||3.3|
|4-card inside Straight||Straight||16.5%||6|
|Any Three-of-a-kind||Full House||24.0%||4|
I have put in a number to use to multiply your proposed bet in order to compare it with the pot to see if you'll be betting with a positive expectation and they're a little on the conservative side, so adjust them if you can live with more risk, especially where you already have a "made" hand, such as Trips, etc. As I explained above, sometimes the hand you're hoping to improve will be good enough to win the pot, so over-betting a little probably won't hurt you in the long run, but remember that 4-card Straights and Flushes are worthless if they don't convert, so I'd advise against "pushing the envelope" when it comes to betting those hands.
As I said in Lesson 1, Internet poker rooms are different than their brick-and-mortar counterparts and the instant tabulation of the pot's value is one of those distinctions. Rather than spending your time trying to figure what's in the pot, you can spend it by seeing if your bet will have a positive EV and, in the long run, that'll be worth a lot of $$$ to you.
Okay, got some homework for you, then that'll do it until next time.
Go to the office game and purchase several hundred dollars of chips. Bring lots of beer and pizza. Your game will also improve if you purchase a nice felt layout and donate it to the game.
Try to play as much as you can, because there's no teacher like experience.
However, before you play, copy the "pot probability" chart presented above and keep it near you so you can use it in your play. Having a calculator nearby is probably also a good idea to get you on the road to playing hands with positive expected value.
For an excellent dissertation on how to perform the math that created the chart above, go here: http://www.math.sfu.ca/~alspach/computations.html
"The end depends upon the beginning." I heard that line in a movie recently and it certainly applies to Hold 'em poker, although that's not what the movie was about. At best, it's extremely difficult to make a comeback if you enter the pot of a Hold 'em game with a bad hand. I see it all the time and it happens, I guess, because so many people feel about poker like they do about Blackjack; "it's all luck, anyway, so what's the difference?" Well, if you've studied my Blackjack lessons the least little bit, you know it's not "all luck" by a long shot. Sure, there is a luck factor that we cannot deny (I prefer to call it "variance") but making the mathematically proper play for each and every hand goes a long way toward reducing the luck factor in Blackjack and that's what we call playing Basic Strategy.
Unfortunately, playing Basic Strategy alone will not give you an edge over the casino - which is why my Blackjack lessons also teach you how to count the cards - but the proper Basic Strategy for a given set of rules in a Blackjack game will reduce the casino's edge over you to a minimum; generally 0.5% or even less.Hold 'em poker also has a "basic strategy" and it begins with the first two cards you're dealt in the game, your "pocket" or "hole" cards. (I suppose that "pocket" cards is more the poker expression, so I'll try to use that when I'm talking about a player's two face-down cards in a Hold 'em game, but forgive me now and then when I lapse into calling them "hole" cards). Anyway, it's easy to imagine that if you were always dealt a pair of "pocket" Aces, you would win tons of $$$ at Hold 'em. Of course, it wouldn't be long before no one would play against you, but you get the idea.
Great cards in the pocket are the start of a great hand. In poker, as in Blackjack, great hands win most of the time. Not all of the time, mind you, just most of the time. We don't always win with a hand of 20 versus a dealer's 6 in Blackjack, nor will we always win with AA ("pocket rockets" in poker slang) in Hold 'em, but it's still a good way to start.
So, how do you make sure you have a good start for a Hold 'em poker hand? Well, that's the beauty of the game of poker. If you don't like your first two cards, you throw them away!
It's somewhat like the surrender rule in Blackjack, except it doesn't cost as much. If you're familiar with surrender, you can stop the play by giving up half your bet and, if surrender is allowed in the casino where you're playing, you should do it whenever the mathematics say you'll win less than 50% of the time. But 50% is a fairly steep price to pay for getting out of the hand. However in poker, it's not nearly that much. In most poker games with 8 to 10 players, you'll have to post a "small blind" and "big blind" bet only about once every 8-10 hands. All of the other hands you'll get cost you nothing to throw away, so in, say, a $10/$20 game with a $5 small blind bet and a $10 big blind bet, it'll cost you only $15 for each "round" of 8-10 hands to toss them. That's a little more that $1.50 per hand and, with a $10 minimum bet per round, the percentage is only 15-20% if you always fold. It would be stupid to always fold, of course, but I want to contrast this with surrender in Blackjack where it would cost you 50% of your total bets if you always did it.
The point I'm trying to make here is that you do not have to play poor cards in a Hold 'em poker game, but most beginners do.
The wise player enters the pot on his or her own terms or s/he simply doesn't play. This takes a certain amount of patience that many beginners seem to lack ("Hell, I'm here to play Hold 'em poker, not Fold 'em poker") and you can take advantage of that. Just as it takes patience for the count in a 6-deck Blackjack game to get into positive territory, so it is with Hold 'em. Good pocket cards don't come along on every deal, so you've got to fold a lot if you expect to make any $$$ from this game.
There's no arguing that the game of Hold 'em poker is more complicated than the game of Blackjack, but both use decks of 52 cards and both are subject to mathematical analysis, so it's actually possible for us to determine which sets of pocket cards are worth playing and which are not.
Let me give you a crystal clear example: Which pocket pair do you think will win more, KK or 22? Hopefully the answer is obvious. A pair of deuces can be beat by any other pair out there but a pair of Kings can only be beaten by a pair of Aces.
Of course, both are beat by two-pair, a set of Trips, etc. so a pair of anything isn't necessarily an automatic winner when all five community cards have been dealt. But it's actually fairly easy to determine which pocket cards will win in the long run and which won't. It's not exactly like determining how much we'll make with a 20 versus a dealer's 6 in Blackjack, because your position at the poker table, the cards that come on the flop, the turn and the river (Unfamiliar with these terms? See Lesson 1.), the other players' cards, how much is in the pot and a variety of other factors will ultimately determine the value of a starting hand. But, believe it or not, we can assign some average values to all of those variables and come up with a nice list of playable pocket cards, which I'll present below.
But before I do that, let me explain my "grand scheme" here. What I intend to ultimately present to you is a Hold 'em Poker Basic Strategy Matrix, which is very much like the matrix I use in teaching Basic Strategy for Blackjack.
But the Hold 'em matrix is going to be a bit more complicated because it will take into consideration your position at the table, the number of players that called the bet before you, any raises, etc. Complicated? Yes. But remember that I'm teaching you how to play Hold 'em poker at online poker rooms, so you won't have to memorize anything! Just print out what I show you and keep it by your computer as you play. Of course, if you are willing to do some memorizing, then the process of evaluating a hand will proceed more quickly, plus you might want to use this information in a brick-and-mortar casino where using a "cheat sheet" may not be appropriate.
Like any other matrix, mine will be built in layers that hopefully have some sort of rhyme and reason about them. But I definitely know where to start and that is to give you a list of the minimum hands you should play.
What I mean by that is this: Your pocket cards can only be one of three types, pairs, suited cards or unsuited cards. Obviously pairs cannot be suited; there is only one King of spades in a deck; get two King of spades as pocket cards and there's a definite problem. Back in the Old West, you'd probably get shot for that. But to continue along, besides pairs (cards of equal "rank" but different suit), you can get suited cards (different rank but same "suit") or unsuited cards (different suit, different rank) and that's it. Within all of those various permutations of cards, there are about 1300 different two-card combinations that can make up the pocket cards in a Hold 'em game.
Play long enough and you'll get all of them, but there are only about 250 or so that you should bet on. Except for the pairs, each set has one card that is higher in rank than the other and that's what forms the basis for my minimum starting hand list. For example, you might be dealt 10c7d (10 of clubs, 7 of diamonds) so the first thing you do is look at the card of the highest rank, which is the 10 of clubs. If the lower card of the two is equal to or higher than the minimum I list, the hand may be played. I say "may" because as we go along, you'll see that your position at the table, the number of raises you may have to call, etc., will all have an effect on whether or not you play the hand. But if the lower card of the two is outside the "minimum", you'll just fold the hand, regardless. So, I guess this isn't so much a list of hands to play as much as a list of hands to not play.
Let me amplify my example with the 10c7d hand. The absolute minimum hand you should play where the 10 is the high card is 10-7s.
This means "10, 7 suited"; in other words, the two are of the same suit, like spades, hearts, diamonds or clubs. Remember that this is the minimum hand, so it's okay to play 10-8s or 10-9s, because they are "above" a 10-7s. What about a 10-Jack, you ask? Well, that falls under the Jacks hands, because we always work off the higher card so don't get confused. Okay, what about 10 and something unsuited? The minimum hand there is 10-8o (10, 8 offsuit). I'm using a small "o" to represent unsuited only because that's the way it's done by most poker writers. I think it should be "u", but they got here before me, so I'll do it their way. Okay, so now we know that the minimum hand with a high card of 10 where the cards are not suited is 10, 8. This means it's okay to play 10-9o, but not 10-7o. The cards would have to be suited for that. Obviously, 10-5, either suited or unsuited is outside the range, so it should never be played, period.
As you go through the list, keep in mind the rationale for most of these choices. Pairs can be improved in many ways and high pairs (Aces-Jacks) can often win on their own. Two suited cards of different rank can win by turning into a Flush, a Straight or a Straight Flush, or by improving to Two-Pair, Trips, etc.
Two pocket cards of different rank and suit are not likely to turn into a Flush, and while they might make Two-pair, Trips, etc., they'll most likely either make a single Pair or, if all goes well, a Straight. Generally you'll see that the "bottom" card is at or near the low end of a Straight Flush for the higher card. For example, the minimum hand for a Queen is Q-8s (Queen, 8 suited) because the 8 is the lowest card that will make a Straight or Straight Flush with a Queen. If the Queen and the other card are not suited, the minimum hand is Q-9o. This makes sense, because you're giving up some "flush power" with this hand; it'll take four cards of whatever suit the Queen is in to make a Flush and somebody else may have the King or Ace and beat you. Just so we're clear on this, if the higher card is a Queen and the lower card is of the same suit, Queen-8 is the minimum hand which means it's okay to play the hand with a suited 9 or 10, also. But if you have, say, a suited 7 (or lower) with the Queen, the hand should be folded. If they're offsuit, then a 9 is as low as you should go; not even an 8 should be played, let alone a 7 or lower.
Make sense? I hope so, but if not, don't hesitate to e-mail me your questions. I always answer my mail personally and I try to do it within 2-3 days at the most. You'll find my address at the end of the lesson. Okay, so here's my list of minimum starting hands. Remember that s = suited and o = offsuit or unsuited. Oh, yeah "x" means any card. By the way, this list is for Limit Hold 'em; No-limit starting hands would be quite a bit different.
Notes and comments:
While it's best to memorize this chart, until you do just print it out and have it near you when you're playing. You can see that as the higher card goes down in rank, the spread between it and the lower card gets tighter. That's mainly because the only hope you have with a starting hand like 7-6o is to make a Straight and more Straights can be made when there are fewer "gaps" to deal with. For those of you who are Video Poker players, you know exactly what I mean; in fact, I found my experience at playing VP very helpful in recognizing playable situations. Now look at the minimum hands for the mighty Ace. If the lower card is of any rank and is suited, the hand is playable, but if it's unsuited, it should be no lower than a 10. As you'll find out, most players will cling to A-xo until the bitter end. And you'll most certainly lose some nice hands to something like A-6o, but in the long run, it shouldn't be played. To draw an analogy to Blackjack one more time, folding A-6o is like hitting A-7 versus a dealer's 9; not everyone does it, even though it's the proper play. It may not "feel" right, but you'll make more $$$ in the long run if you'll do it. Math does not have room for "feelings". Cold, perhaps but that's how it is.Pairs: No pairs are listed on here because all of them are playable at one time or another. Just remember that this list (and the pairs) is not a license to play these hands at any time, under any set of circumstances. For example, you'd be crazy to play 10-7s in an early position after 3 players have raised behind you. As I said earlier, this list is as much about what not to play as it is about what's playable. So stop calling with those Q-3s hands and be patient until I show you the entire matrix. That will incorporate this list and the pairs into a complete strategy that takes into consideration your position, how many bets you have to call and so on. In the meantime, I have some homework for you and that'll wrap it up.
A lot of the work that goes into deciding which hands are worth playing and which are not is derived from simulation. While a "sim" is not as precise as strict mathematical anaysis, if the simulation encompasses a large number of hands, it can come very close. I do these sims on my copy of "Turbo Hold 'em" by Wilson software, which I mentioned in a previous lesson. For a look at a sim that was performed by others to determine the expected value of many starting hands and the best way to play them in early position ("under the gun" or first to bet), go here: http://www.posev.com/poker/holdem/sim/utg10ta03b.txt/Print out the List of Minimum Starting Hands, right now while you're thinking about it and place it by your computer so you can refer to it as you play. Go play some limit Hold 'em for funny-money at one of our recommended poker rooms (InterCasino Poker is really good for limit games) and use the Minimum Starting Hand List as you do so. If you'll kindly click on one of their ads here, it'll help keep this site free for you. We appreciate it.
- Here's another terrific poker site to visit: http://www.cardplayer.com/ Just don't click on any of their ads, save all your clicking for here ;-). It's the Website of what is probably the best poker magazine being published today.