Mathematics of NLHE Ep 1

Covered in this episode:

• Probability and Odds
• Variance
• Rule of 2 / Rule of 4
• Pot Equity
• Expected Value (EV)

Math is not a replacement for physical tells, timing tells, our gut feelings, and our hand reading. Math is the tool we use to determine what the best course of action is based on the information we gathered through the tells, our gut, and our logical hand reading deductions.

Converting a Ratio to a Percentage

• Convert 2:1 to percentage
• English: If you are a 2:1 favorite, that means you expect to win 2 times for every 1 time you lose. That means you are winning 2 out of 3 or 2/3 of the time 2 divided by 3 is .666666666 therefore you haave a 66% chance of winning.
• Math: Add both the "left side" and the "right side" together and divide it by the number of our "wins".
• (wins)/(left side + right side)
• Ex 1. - 2:1 favorite, you expect to win 2 times for every 1 time you lose
• 2/(1+2)
• 2/3 = .666666666 or 66%
• Ex 2. - 3:2 favorite, you expect to win 3 times for every 2 times you lose
• 3/(3+2)
• 3/5 = .6 or 60%
• Ex 3. - 4:1 dog, you expect to win 1 time for every 4 times you lose
• 1/(1+4)
• 1/5 = .2 or 20%

Converting a Percentage to a Ratio

• With a ratio, we put "wins" on the left side and "losses" on the right
• Ex 1. - If we're 75% to win that means out of 100 tries, we win 75 and lose the other 25.
• 75% = 75:25 or 3:1 favorite

• Ex 2. - 33%
• 33:67 or 2:1 dog

• Ex 3. - 83%
• 83:17 or 4.9:1 favorite

Variance

• Mathematical Definition: Variance is a measure of statistical dispersion
• Example of a "High Variance Play"
• You have QJo and Board is AK95r, pot is $110. Opponent bets $10.
• As we'll find out, you are getting the correct odds to call here, however, you will only hit your gutshot straight draw about 1 time in 11. You could miss that draw many, many times in a row before you finally hit and get paid off for all the calls.

Odds to Memorize

• Preflop AI matches:
• Pair over pair (AA vs TT) - 4:1 or 80%
• Pair vs two overcards (88 vs AK) - 1.2:1 or 55%
• Pair vs two undercards (KK vs 98) - 4.9:1 or 83%
• Pair vs higher card/lower card (QQ vs KT) - 2.3:1 or 70%
• Two higher cards vs two lower cards (KQ vs 98) - 1.9:1 or 65%
• High card/low card vs non pair (AT vs KQ) - 1.2:1 or 55%

• Postflop:
• Hand - Outs - Flop - Turn
• Pocket pair (postflop) - 2 - 10.9:1 or 8.4% - 22.3:1 or 4.3%
• Gutshot straight draw - 4 - 5.1:1 or 16.5% - 10.5:1 or 8.7%
• Overcards - 6 - 3.1:1 or 24% - 6.7:1 or 13%
• OESD - 8 - 2.2:1 or 31.5% - 4.7:1 or 17.4%
• FD - 9 - 1.9:1 or 35% - 4.1:1 or 19.6%
• FD + overcard - 12 - 1.2:1 or 45% - 2.8:1 or 26.1%
• FD + OESD - 15 - 1:.95 fav or 51.2% - 2.3:1 or 30.4%
• FD + overs - 15 - 1:.95 fav or 51.2% - 2.3:1 or 30.4%

Rules of 2 and 4

• Rule of 4: on the flop, you can multiply your outs by 4 and that will give you the approximate percentage chance you'll draw to the best hand.
• Rule of 2: on the turn, you can multiply your outs by 2 and that will give you the approximate percentage chance you'll draw to the best hand.
• These rules are very rough estimates and it tends to work better the fewer outs you have.
• Rule of 4, Solomon: (# of outs)(4) - (# of outs - 8) for outs more than 8.
• Ex. 12 outs. 12 x 4 = 48. 12 - 8 = 4. 48 - 4 = 44%

Hand Ranges

• Definition: A hand range is the collection of every 2 card starting hand that you can have based on the actions taken place throughout a hand.

Pot Equity

• Pot equity = (out % chance of winning) x (the pot)
• Equity can only be calculated when we know the opponent's hand or have an idea about his hand range.
• Ex. There is $500 in the pot and we have 25% "equity". $500 x .25 = $125

Expected Value (EV)

• Definition: EV is the determination of what our long term expected outcome is on a given hand in profit or loss.
• EV is also known as "Sklansky Bucks," named after renowned 2+2 poker author David Sklansky
• Ex. 1
• We have AcKc and $1000 effective stacks. Our opponent holds 8s7s on a AhQs2s flop. We bet $175 into a pot of $250 and our opponent shoves for his remaining $875. What is the EV of our call?
• Our equity in the pot or our chance to win this hand is about 63%
• total pot = (250+175+875) = 1300 and it is 700 for us to call
• EV = .63(1300) - .37(700)
• EV = $560

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A Poker Player's Command Center

Not a post on poker strategy but has good advice on products that will improve productivity and health when sitting for long periods in front of a computer.

http://dailyvariance.com/slowhabit-blog/106-a-poker-players-command-center.html

Poker Timing

http://dailyvariance.com/slowhabit-blog/66-poker-timing.html

Despite its importance, timing is a concept that is rarely discussed in forums. It is probably because players don't know about it or underestimate its importance.

When you see a great player check-raise all-in on a flop of 3s 5d Jc holding As4s and win, the first thing that comes into your head is, “wow, he is a great player and he is currently playing great so that play must have some +EV attached to it.” Naturally, you try to implement it in your game, only when you do it, your stacks move themselves over to your opponent's side. The reason is your timing is off. You don't know when to make the play. In other words, you are right at the wrong time.

This is what a great player so great. He KNOWS when he should do it. A simple analysis such as, “he has been running me over. Therefore, I have to fight back somehow and this is a decent hand to do it with.” Using such logic is *kind of* correct. The problem is if your opponent is any good, he is anticipating such a play from you as well.

So what should one do to improve one's timing? The answer is experience. There is no guideline nor formula to improve one's timing. It is based on your instinct and your instinct is developed over the years by trusting your reads.

The point of this writing is to help you realize such a concept exists so you won't check-shoving with As4s on a 3s 5d Jc in any re-raise pots and wonder why certain players run so good.

By Tri "Slowhabit" Nguyen

I Play Solid

http://dailyvariance.com/slowhabit-blog/32-i-play-solid.html

... But I Can't Seem To Win.

I hear a lot of players say this and it gets funnier every time I hear it. The reason is, they don't play solid. What they meant is they play in a robotic manner and hope to cooler someone. So when they win these big cooler pots, they play well and run well. If they lose, they run bad. There is some truth to this because you obviously run bad if you don't win the cooler pots.

But poker is much more than those big cooler pots. Poker is a game that is defined by better players accumulating small edges here and there. When you sit and play a solid game, better players are taking a more active role in maximizing their EV. One example, they play with a little more aggression. They fight extra hard for pots. They think of opponents' ranges before betting. They dictate the flow of the game and is always aware of their images.

People usually say they play a good game and they are just running bad. But whenever we have sweating sessions, there are so many small things they miss out. Three notable examples are cbetting too much, neglecting the small pots, and passive pre-flop play. It's hard to detect these problems because if you knew it, you would have fixed it already. When we look over our sessions, we usually focus on the big pots that we win/lose but neglect to think about our overall game and what we can do to improve it. A player who is active pre-flop will take a lot of uncontested pots and thus, don't have to win those big cooler pots to have a winning session. It's amazing how Doyle was sooo ahead of his time.

Next time when you're at the table, remember to think throughly about a situation before clicking a button. Don't be lazy. Talk aloud. Practice. Perhaps you'll realize that maybe, just maybe, you weren't as solid as you assumed.

By Tri "Slowhabit" Nguyen

Baluga Whale Theorem

http://archives1.twoplustwo.com/showflat.php?Cat=0&Board=ssplnlpoker&Number=6605819&fpart=1&PHPSESSID=

You (100bb) have Ad Kc in MP at a 6max game. UTG (100bb) limps, you raise to 5xBB, Button (100bb) calls, UTG calls.
Three to the flop (pot ~15bb), which is:
Ac 4s 6s
UTG checks, you lead out for 12bb, Button folds, UTG calls.
On to the turn (pot~ 36bb)
8h UTG checks, you bet 25bb, he raises all-in.
We fold. One pair is not good here. A draw does this about never. We have to bet this turn because we can't let spades draw, and we need value from worse A's, but now that he raises, we can rule out worse A's and draws.
However, to slightly alter Isura's example, lets say we have Ac As, same pf action as before and the flop (pot~15bb) comes down:
Kc 2h 2d UTG checks, we bet 12bb, Button calls, UTG folds.
the turn (pot ~36bb) comes:
Jd We bet 25bb, Button raises all in.
Here, I think we should call. We are very likely to see KJ here, as well as AK or KQ. Occasionally we see a random 2, but we are ahead of his range more often than not.
Basically, the whole point of the "Baluga theorem", as I see it, is to strongly reevaluate one-pair hands facing a turn raise.

By BalugaWhale

COTW #7 Review/Analysis

This is a review of COTW#7: http://forumserver.twoplustwo.com/78/micro-stakes-full-ring/concept-week-7-3betting-436868/

I’m not a big fan of this particular cotw. The HH examples were played poorly and the OP does not explain what ranges to 3-bet with. Our 3-betting range is often going to either be for value or polarized depending on the opponent and that will require that we know how s/he plays to some degree. Knowing why we 3-bet is only half the battle and know what to 3-bet with is going to help us create a much better strategy as far as post-flop play goes.

The 87s hand was also played poorly on all streets. Pre-flop I think isolating here with the intention of calling a 3-bet with 87s is going to be pretty bad even if u have good implied odds. The villains range is so tight that it will be difficult for the Hero to have much FE which is mainly what SCs are best for. Hero will not hit the flop hard enough often enough to make it worth his iso/call 3-bet investment. Depending on table dynamics, I think flatting can be ok as well as raising smaller to call the min 3-bet if u want the pot heads up. Post-flop play was also very meh. There is very little value in calling the flop because the villain is not very likely to double barrel on this type of board with a hand like AK and by calling, you give him a chance to catch up. With the amount of equity we have, shoving is going to be more profitable as we can never fold the flop and we have some FE if the villain is nitty enough. Again, all this is going to be opponent specific, but for the most part I think raising is the better play. The turn bet was way too big considering the spr. Against the villain’s range, we practically have the nuts here. Our huge bet is not optimal for value and accomplishes nothing except for letting our opponent play perfectly against us. We can make a smaller bet here for value and it will help us balance the times we decide to double barrel bluff the turn for cheaper.

Not sure about the JJ hand or what it teaches. This post is meant for fullring players and as such, the hand was played at a 9 handed table. Folding JJ IP to a 3-bet seems a bit nitty but is probably correct against other FR nits, especially one from the blinds against your UTG open.

The 66 hand that the OP 3-bets with is explained poorly. He runs an equity calculation of his hand versus the villains opening range but what is important is not so much the villain’s opening range but his 3-bet calling range which is generally going to be tighter and stronger. His 3-bet there with 66 is a total bluff unless he expects the villain to call with his whole opening range and even then he is 50/50.

The 42s hand requires more explanation. He is most likely 3-betting a polarized range here since villain is folding so much but OP does not state that. He only says you should be asking yourself why, which is not going to be enough. You need a coherent game plan pre-flop that also extends to post-flop play.

Not sure about the Q9s hand. Seems like a spew to 3-bet it IP when the hand has so much post-flop potential. Can be profitable if villain is folding a lot to 3-bets and/or folding to c-bets a lot post-flop. His 3-bet sizing seems to be a tad big as well. He is creating a spot where there will be a ton of RIO when he is called with a fairly big pot/shallow spr.

The HHs were not the best, but there are still some pretty good content in the post and worth a read. As is with everything you read, question it and don’t take it to heart. Everyone’s game is a little different and what you take away from this post may be different from what another takes away. Nothing in poker is set in stone.