Poker bankroll management, building and strategy – online poker guide

Angular Momentum Conservation in live poker

Monday, December 22nd, 2008 | Poker Articles, Poker Mathematics | 3 Comments

Submitted by Rakewell, this article belongs to the Poker Mathematics series.

Whoever said poker players are superstitious? The article below written by Rakewell treats the dealing and receiving of cards when playing live poker from a purely natural scientific viewpoint. It’s all about angular momentum conservation baby:-)!!! Enjoy……

I’ve decided to break down and share with my readers an important poker principle that you won’t find in any book or DVD or training video currently on the market. That’s because there are some things that the pros keep to themselves in order not to give up all of their edge to the general public.

You’re sitting in a casino poker room and the dealer pitches you a card. It lands nicely right in front of you. But you see that it is not oriented the way you’d like in order to be able to peek and see what it is. Personally, I like them with the long axis pointed toward/away from me, but others like them with the long axis oriented right/left. Either way is fine, but whichever way you prefer, you have a problem when the card lands close to 90 degrees from the way you need it.

The dilemma is this: You have to rotate the card(s) either clockwise or counterclockwise–but which?

Amateur players often assume that it couldn’t make any difference. After all, what is printed on the face of the card can’t change because of how you turn it. “Preposterous,” these people would say.

But it does matter. You absolutely must apply the rotation in the same direction that the card was spinning as it arrived in front of you (which usually depends on whether the dealer is right-handed or left-handed). If you rotate it in the opposite direction, terrible things happen. The cards get dizzy from the sudden change. This is especially true with the face cards. If you upset the delicate equilibrium of, say, a queen, do you think she is going to call out to her peers to come join her in this hand? No! She’s upset. She’s nauseated. Her inner ear thing is all out of whack. She’s going to just sit there and try to recover. She might even throw up a little. By the time she’s feeling better, the hand is over and you’ve got nothing except a little spot of queen vomit.

The underlying mechanism is different for the non-face cards. It’s not a dizziness problem, but one of conservation of luck, which is closely related to the conservation of angular momentum. Do you remember a carnival ride when you were a kid, in which you stand against a round wall, and they start spinning it, and after it’s really going they drop the floor out from under your feet and you stick against the wall by centrifugal force? It’s the same with the cards. You need to keep them spinning in the same direction that the spinning was initiated by the dealer. If you suddenly reverse it, the luck all falls out, in precisely the same way that you would have fallen down into the pit of that ride if they had suddenly thrown it into a reverse spin.

The mathematics of this has actually been worked out in some detail by the boys at Lawrence Livermore Laboratory. It’s beyond the scope of this blog, but trust me on this. Or go to your local library, ask where they keep the back issues of The Journal of the American Society of Theoretical and Applied Serendipity, and look it up for yourself.

You have to treat the cards with respect, and that includes not jarring them into a sudden reverse spin. Once you think about the underlying mechanisms, it’s rather obvious, isn’t it?

So now you know.

Check out Rakewell’s Poker Grump blog

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Tags: Poker Articles, Poker Mathematics

Poker ev; calculating the winning plays

Saturday, November 8th, 2008 | Poker Articles, Poker Mathematics | 9 Comments

This article is a part of the Poker Mathematics series.

This article is number 3 in a series of 3 articles covering the (in my opinion) most important mathematical aspects of poker:

• Calculating pot odds: see Poker pot odds; all you need to know
• Calculating poker probabilities: see Poker probabilities; all you need to know
• Calculating expected value: see +EV poker; making the winning plays

Having explained how to calculate poker pot odds and poker probabilities in my two previous articles we now move on to applying these concepts to improve your poker game by making the winning plays every time. The concept you will need to learn is EV, which is short for expected value.

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In poker, EV is a measure of how much you will be payed back on average on a 1$ bet:

• EV < 1 – If you always play hands with an EV of less than 1 you will lose on average
• EV = 1 – If you always play hands with an EV equal to 1 you will break even on average
• EV > 1 – If you always play hands with an EV of more than 1 you will win on average

Needless to say it should be the goal of any poker player to make plays that always belong to the EV > 1 category.

When it comes to calculating your EV for any given poker hand that the European decimal odds system, which I favor, excels.

Simply multiply the decimal pot odds you are given by the probability that you will win the hand, and you have your EV. Simple as that.

When using the European decimal odds system it is also easy to calculate either the pot odds or probability needed to break even (EV = 1) from the following relationships:

• pot odds = 1/Probability
• probability = 1/pot odds

Here are some examples of how these calculations work in real situations:

• You have 15 outs on the turn to win the hand and you have to call 700 into a 1500 pot. What is your EV for the given situation? Well the probability for winning the hand is (15*2+1)% = 31% = 0,31 using the easy rule of thumb. Your decimal pot odds are (1500+700)/700 = 3,14. This gives an EV of 3,14*0,31 = 0,97. Therefore you will on average lose 3 cents for every dollar you bet by making this call.
• In the example from above, what decimal pot odds do you need to break even? The answer is 1/0,31 = 3,23.
• You are holding a pair of eights and the flop is high cards. To win the hand you need to hit a set on either the turn or the river. What pot odds do you need on the turn and the river in order to break even? Again using the easy rule of thumb the probability of hitting your set on the turn is (2*2+1)% = 0,05, so you need a pot odds of 1/0,05 = 20 in order to break even. The same applies to the river.

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Even though these calculations are straightforward you will need some practice to be able to perform them fast enough at the poker tables. Therefore you might want to make a set of guidelines to memorize. The list below will get you started:

• If your single opponent bets the pot on the flop your pot odds are 3 for calling to see the turn card. Therefore you need a probability better than 33% to make the call and win in the long run. This corresponds to approximately 16 outs.
• If your single opponent bets half the pot on the flop your pot odds are 4 for calling to see the turn card. Therefore you need a probability better than 25% to make the call and win in the long run. This corresponds to approximately 12 outs.
• When you flop a flush draw and want to see a turn card you need pot odds 5,2 to break even.

For those of you using the fractional odds system I would recommend that you convert your pot odds to the decimal system when calculating your EV values. For conversion between the European decimal odds system and the UK fractional odds system see poker pot odds; all you need to know.

Having now covered the basics of the poker mathematics essentials I will move on to the more advanced concepts of implied and reverse implied odds in later articles. The combination of a sound understanding of poker mathematics with bankroll management for poker should give you an edge against many opponents at lower limits.

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Tags: Poker Articles, Poker Mathematics

Poker probabilities; all you need to know

Sunday, November 2nd, 2008 | Poker Articles, Poker Mathematics | 5 Comments

This article is a part of the Poker Mathematics series.

This article is number 2 in a series of 3 articles covering the (in my opinion) most important mathematical aspects of poker:

• Calculating pot odds: see Poker pot odds; all you need to know
• Calculating poker probabilities: see Poker probabilities; all you need to know
• Calculating expected value: see +EV poker; making the winning plays

Calculating exact poker probabilities for each hand you play can be a complex mathematical operation which is simply not feasible with the short response times in online, and to a lesser extent live poker. Poker players have different approaches to deal with this challenge. Some players use poker tools in the form of programs that continuously calculate and display probabilities and pot odds. Others memorize the probabilities of a wide range of poker situations. There are  players who don’t calculate exact probabilities but instead base their decisions on previous experiences and intuition (I myself belong to this group, but am motivated to include more mathematical considerations in my game). In this article I will explain how exact poker probabilities are calculated from the number of outs you have and also present a very useful shortcut to calculating poker probabilities from outs.

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First of all a clear definition of a poker out is needed: If you believe your hand needs improving after the flop to win the pot, then an out is a card that will do just that. Say for example that you are holding AK suited and the flop gives you two of your suit. Your opponent has QQ (none in your suit), so you need to improve your hand in order to win the pot. Your outs would be any card of your suit (9 remaining in the deck) or any A or K (6 remaining in the deck) giving a total of 15 outs. Keep in mind that counting your outs is always an estimate. If your opponent has flopped a monster what you might consider outs will end up costing you your stack. On the other hand your opponent could also be bluffing in which case your AK might already be the best hand.

Once you have estimated how many outs you have, calculating the probability of either the turn or the river being one of your outs is fairly straightforward. Simply divide the number of outs with the remaining number of unseen cards in the deck. Using the example from above the probability of the turn (P(turn)) being one of your suit, an Ace or a King is 15 divided by 47 (52 card minus your two hole cards and the three cards constituting the flop) or roughly 32%. If one of your outs did not come on the turn and the board did not improve your opponents QQ you still have 15 outs on the river giving you a probability of 15/46 or  roughly 33% (P(river)) that one of your outs will come on the river.

Calculating the probability that one of your outs from the example above will come on either the turn or the river (P(turn or river)) is a bit more tricky. This probability is NOT simply the sum of P(turn) and P(river) because by summing these two probabilities you will be including outcomes that are not possible. For example if the Ace of Clubs hits on the turn it cannot come on the river. In mathematical terms the events of one of your outs coming on the turn or the river are not independent. I will give you two ways of calculating the exact probability of one of your outs coming on either the turn or the river.

Method 1:

There are 3 possible outcomes of hitting on of your outs on the turn or the river, namely you hit one on the turn and not on the river (P(turn not river)), you don’t hit one on the turn but one comes on the river (P(river not turn)) and you hit one on both the turn and the river (P(turn and river)). The probabilities are calculated as follows:

P(turn not river)= P(an out comes on the turn)*P(an out does not come on the river after an out has come on the turn) = (15/47) * (46-14)/46 = 0,222

P(river not turn)= P(an out comes on the river)*P(an out does not come on the turn) = (15/46) * (47-15)/47 = 0,222

P(turn and river)= P(an out comes on the turn)*P(an out comes on the river after an out has come on the turn) = (15/47)*(14/46) = 0,0971

Summing the 3 probabilities from above gives 0,54 which is the probability that one of your 15 outs will hit on either the turn or the river

Method 2:

The probability of hitting one of your 15 outs on either the turn or the river is the complement of the probability not hitting one of your outs on both the turn and the river:

P(turn or river) = 1- P(an out does not come on the turn)*P(an out does not come on the river) = 1-((47-15)/47))*((46-15)/46)) = 0,54

I hope you agree with me that none of the methods above are very practical when it comes to calculating poker probabilities when you are sitting at the poker tables. Luckily there is an easy to remember rule of thumb that does a good and quick job of calculating poker probabilities from poker outs.

Easy rule of thumb:

P(turn or river) = (4*(number of outs) – 1)%

P(turn) = (2*(number of outs) +1)%

P(river) = (2*(number of outs) +1)%

Taking our 15 out example from above, the easy rule of thumb yields the following probabilities:

P(turn or river) = 59% = 0,59

P(turn) = 31% = 0,31

P(river) = 31% = 0,31

which is accurate enough for most purposes.

I will leave it to you to practise estimating how many outs you have in any given poker hand since this is one the most important disciplines in poker. The list below should get you started:

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Outs when you are drawing to:

• Three of a kind: 2
• High Pair: 3
• Full house when you have hit two pair: 4
• Open-ended straight: 8
• Flush: 9
• Flush or high pair: 12
• Flush or pair: 15
• Open-ended straight, flush or pair: 21

In my next article I will explain how to use the concepts of poker pot odds and poker probabilities to become a winning poker player in the long run.

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Tags: Poker Articles, Poker Mathematics

Poker pot odds; all you need to know

Saturday, October 25th, 2008 | Poker Articles, Poker Mathematics | 10 Comments

This article is a part of the Poker Mathematics series.

This article is number 1 in a planned series of 3 articles covering the (in my opinion) most important mathematical aspects of poker:

• Calculating pot odds: see Poker pot odds; all you need to know
• Calculating poker probabilities: see Poker probabilities; all you need to know
• Calculating expected value: see +EV poker; making the winning plays

I have a feeling that many poker players, both experienced and beginners, could do with a brush-up on how to calculate poker pot odds. In this article I will try to explain the concepts so they make sense. Hope I succeed:-)

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Firstly when it comes to odds calculations there are basically 3 different systems:

• The Fractional Odds System favored among bookmakers in the UK. In the fractional odds system you will see odds displayed as fractions such as 6/1 or more commonly 6:1
• The European Decimal System used mostly by bookmakers in Europe (no surprises here). The European decimal system displays odds like decimal numbers such as 1,3, 2,7, 6 etc.
• The American Moneyline System favored by bookmakers in the US, hence the name of the system. In the American Moneyline System you will find odds like +500, -123, +345 etc.

I favor the European Decimal System (I’m European what can I say?) but a quick glance at the available poker literature out there will quickly convince you that the fractional odds system is favored among most poker writers (including my guru Dan Harrington). Luckily this article will give you an idea of how poker pot odds are calculated in all three systems.

Moving on the actual math: let’s say you are playing an online tournament and the pot after the flop is 2500. Your only remaining opponent bets 500, what are then your pot odds for calling? With your opponent’s bet of 500 the pot is now 3000 and you have to call 500 to stay in the pot. Here’s how you would calculate the pot odds for calling in each of the odds systems described about:

• In the Fractional Odds System the fraction quotes how much profit the bettor will make relative to his stake. A fractional odds of 3:1 therefore means that the bettor will make a 300$ profit for every 100$ staked. A fractional odds of 1:3 on the other hand means that the bettor will make a 33$ (100$*1/3) profit for every 100$ staked. In both cases the bettor will receive his original stake back. This means that in the 3:1 example the bettor will receive 400$ (300$+100$) in return and in the 1:3 example the bettor will receive 133$ (100$+33$) in return. Returning to the poker example from above, by calling your opponent’s bet of 500 you could potentially win a pot of 3000. A bet of 500 with a profit of 3000 gives a fractional pot odds of 6:1 (3000/500). As written earlier your initial stake is returned to you along with your profit which means that you in total will receive 3500.
• The European decimal system differs from the other two systems in as much as the bettor in effect hands over his stake to the bookmaker. The decimal odds then reflects the total amount that will be returned to the bettor; that is both his stake and his profit. Let’s see how this works in the 3:1 and 1:3 examples from above. When staking 100$ to win 300$ the bettor parts with his stake and is paid back his stake plus his profit which equals 400$ (100$+400$). In this case his decimal odds are quoted as 4 (400$/100$). In the 1:3 example, the decimal odds are 1,33 (133$/100$). Revisiting the poker pot odds example, the decimal poker pot odds are 7 (3500/500).
• In the American Moneyline system the odds are either positive or negative. The fractional odds of 3:1 from the example above would be written as +300 indicating the amount of money to be won on a 100$ wager. The 1:3 fractional odds would be written as -300 indicating that you would have to wager 300$ to win 100$. Going back to the poker pot odds example the 6:1 pot odds therefore corresponds to +600 in the American moneyline system. As a concluding remark I do not think the American moneyline system is well suited for calculating poker pot odds, so if I were you I would focus on getting familiar with either the fractional or the decimal odds system.

Given the explanations above, converting poker pot odds between the three systems is pretty straightforward. I have listed some examples below. Fill in the blanks yourself to practice converting odds.

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• Fractional odds to decimal odds: write the fraction as its corresponding decimal number and then add one. Thus 8:1 corresponds to 9, 1:4 corresponds to 1,25, 1:8 corresponds to 1,125 and so on.
• Fractional odds to moneyline odds: if the odds is more than even (i.e the decimal odds is larger than 2), convert the fraction to its decimal number and multiply by 100. If the odds is less than even invert the fraction, convert to its decimal number and multiply by -100. For example 7:1 becomes +700, 2:1 becomes +200, 1/4 becomes -400, 1/9 becomes -900 and so on.
• Decimal odds to fractional odds:
• Decimal odds to moneyline odds:
• Moneyline odds to decimal odds:
• Moneyline odds to fractional odds:

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Tags: Poker Articles, Poker Mathematics

Poker Mathematics

Wednesday, October 8th, 2008 | Poker Mathematics | 2 Comments

This article is a part of the Poker Mathematics series.

Being a good enough poker player to generate money playing online is about mastering several aspects of the game. The rules of Texas Holdem (or even Omaha) are deceitfully simple. From learning how to play the game to becoming a pro there’s a lifetime’s learning of a distance though.

In order to master the game, a player needs to be in complete control of its mathematical aspect, as well as its psychology and strategy sides. Mathematics is a very important part of being successful in poker and it does give beginners a pretty good guidance regarding the quintessential question in poker: should I call, should I fold or should I raise?

As soon as you sit down to the table and you realize where you are in relation to the dealer button, mathematics floats into the picture.

The mathematical odds that you get for your starting hands are deeply influenced by your position at the table. If you are under the gun or in a different early position, you should tighten your starting hand selection because the odds that you get for a K, J for instance are not the same as the ones you get for the same hand on the button or in the cut-off.

Starting hand selection is about math as well. Not all starting hands were created equal as some stand a higher mathematical chance to win than others. Correct starting hand assessment takes position into account, but pot odds need to be considered too.

Aggressive preflop play is about mathematics as well. Its aim is to drive out as many players as possible or to make them call and fold on the flop, thus leaving dead money in the pot.

The fewer players there are in a hand (the more you manage to chase away through aggressive preflop betting), the better the odds will be for every single hand that remains in the game. Players who leave dead money in the pot alter the pot odds, and thus they act to increase the odds for those who remain in the game.

Making the correct call has a very clearly defined mathematical side to it. In order to find out whether or not one needs to call a raise, to fold a hand or even re-raise, all he/she needs to do is to compare the odds of making a potentially victorious hand, and the pot odds. The pot odds are easy to calculate.

All our player needs to do is to compare the number of chips in the pot with the number of chips it takes to make the call. If there are $50 in the pot and the bet he’s facing is a $10 one, the odds are 5-1.

He needs to calculate the odds of making his hand (let’s say he has a 4-card flush on the flop). Taking all the cards already on the table into account, as well as the remaining outs, the odds of hitting a flush on the turn are 4.22-1 against. Since this number is smaller than the pot odds themselves, it will make sense for our player to make this call, but not a re-raise, since that would ruin the pot odds for him.

Mathematics can give you a clue or two about what calls you need to make, however, a large part of poker is about psychology and about putting opponents on hand-ranges based on their behavior and betting patterns. That’s where the mathematical element blends into the inexact science of psychology and behavior assessment.

Another wonderful example of how mathematics works in poker is rakeback. In real money play, you pay a rake on every single hand to the poker room. That is how the latter makes its revenue.

Rakeback gives you a certain percentage of that rake back.

Using simple mathematics one can easily prove that sign-up and other such limited-validity bonuses are in fact the equivalent of a rakeback deal, only they carry the rather overwhelming disadvantage that they expire over time.

Rakeback never expires, and if you really want to you can use mathematics to prove how it affects the pot odds you get on every single real money hand you ever play.

Post By Steve.

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Tags: Poker Mathematics