The Mathematics of Poker (2 Viewers)

[23slam]

Man, there's a lot of math in poker. A lot of other factors too. I don't even know what outs are. You guys are all way better than I am. When they check, I bet. When they bet, I raise. If they reraise, I either tank (for show)/fold, or tank/ship.

Actually, the topics here (original topic plus others--like implied odds, and mistakes) are the factors that make no-limit a MUCH better game than limit hold 'em.

 

[Chicken Rob]

Man, there's a lot of math in poker. A lot of other factors too. I don't even know what outs are. You guys are all way better than I am. When they check, I bet. When they bet, I raise. If they reraise, I either tank (for show)/fold, or tank/ship.

Actually, the topics here (original topic plus others--like implied odds, and mistakes) are the factors that make no-limit a MUCH better game than limit hold 'em.

Limit is a much better game in my experience. I miss access to good limit games. Things like mistakes and implied odds are critical in limit.

If you have never played above 5/10, I can understand why you think limit is less good, but I'd rather play a loose aggressive 15/30 or 30/60 limit game, especially 5-6 handed, than any NLHE game of any size or stakes.

 

[abby99]

The math involves grossing up the pot so that the equity percentage applied to the pot including our bet produces the amount of our bet. The formula that I use is ((Pot / (1 - probability of event not happening) - Pot). Admittedly, that's a little unwieldy to perform at the table. One can memorize a few benchmarks, such as:

Pot-sized bet (PSB) gives our opponent 2:1 to call, equity 1/3 or 33%, approx. 16 outs.
Half of a PSB gives our opponent 3:1 to call, equity 1/4 or 25%, approx. 12 outs.
One-third of a PSB gives our opponent 4:1 to call, equity 1/5 or 20%, approx. 10 outs.
One-fourth of a PSB gives our opponent 5:1 to call, equity 1/6 or 17%, approx. 8 outs.

To calculate this on the fly, say we want to bet 2/3 of a $100 pot, or $67. After we bet, the pot will be $167, and we can calculate that $67/$167 is 40%. Divide that by 2 (from the rule of 2 and 4) to get 20 outs. The odds corresponding to 40% are 4 in 10, which is the same as 6:4, which simplifies to 3:2 (i.e 3 to 2). But wait -- we certainly don't want to get out our calculator at the table, so we need to think of this differently. The new pot will be 2/3 + 3/3, or 5/3. 2/3 divided by 5/3 is the same as 2/5, or 40%, which again gets us to 20 outs. Working this backwards from 20 outs to the amount of the bet, take 20 x 2 to get 40% equity. The odds ratio will be 60:40, which simplifies further to 3:2. Then we flip the odds ratio 3:2 to produce the fraction 2/3 and we bet 2/3 of the pot, or $67. Easy peasy, right?

@chkmte's rule of 3 and 6 is a nice shortcut that is dead on at 14 outs; this is the point where the bet is equal to 3x the number of outs. As the number of outs increases or decreases from 14, the difference between the calculated raise and the results from the rule of 3 and 6 increases (check out the attached table and chart). However, the range between 8 outs and 16 outs is consistently within 4 percentage points. As the saying goes, it's close enough for government work.

All of my calculations are based on action between the flop and the turn (the "2" in the rule of 2 and 4).

Questions and corrections are welcomed.

Edit: Updated the attachment.

 

[chkmte]

The math involves grossing up the pot so that the equity percentage applied to the pot including our bet produces the amount of our bet. The formula that I use is ((Pot / (1 - probability of event not happening) - Pot). Admittedly, that's a little unwieldy to perform at the table. One can memorize a few benchmarks, such as:

Pot-sized bet (PSB) gives our opponent 2:1 to call, equity 1/3 or 33%, approx. 16 outs.
Half of a PSB gives our opponent 3:1 to call, equity 1/4 or 25%, approx. 12 outs.
One-third of a PSB gives our opponent 4:1 to call, equity 1/5 or 20%, approx. 10 outs.
One-fourth of a PSB gives our opponent 5:1 to call, equity 1/6 or 17%, approx. 8 outs.

To calculate this on the fly, say we want to bet 2/3 of a $100 pot, or $67. After we bet, the pot will be $167, and we can calculate that $67/$167 is 40%. Divide that by 2 (from the rule of 2 and 4) to get 20 outs. The odds corresponding to 40% are 4 in 10, which is the same as 6:4, which simplifies to 3:2 (i.e 3 to 2). But wait -- we certainly don't want to get out our calculator at the table, so we need to think of this differently. The new pot will be 2/3 + 3/3, or 5/3. 2/3 divided by 5/3 is the same as 2/5, or 40%, which again gets us to 20 outs. Working this backwards from 20 outs to the amount of the bet, take 20 x 2 to get 40% equity. The odds ratio will be 60:40, which simplifies further to 3:2. Then we flip the odds ratio 3:2 to produce the fraction 2/3 and we bet 2/3 of the pot, or $67. Easy peasy, right?

@chkmte's rule of 3 and 6 is a nice shortcut that is dead on at 14 outs; this is the point where the bet is equal to 3x the number of outs. As the number of outs increases or decreases from 14, the difference between the calculated raise and the results from the rule of 3 and 6 increases (check out the attached table and chart). However, the range between 8 outs and 16 outs is consistently within 4 percentage points. As the saying goes, it's close enough for government work.

All of my calculations are based on action between the flop and the turn (the "2" in the rule of 2 and 4).

Questions and corrections are welcomed.

View attachment 47323

Gentlemen, you lose.

 

[abby99]

Limit is a much better game in my experience. I miss access to good limit games. Things like mistakes and implied odds are critical in limit.

If you have never played above 5/10, I can understand why you think limit is less good, but I'd rather play a loose aggressive 15/30 or 30/60 limit game, especially 5-6 handed, than any NLHE game of any size or stakes.

For some, comparing NL/PL to fixed limit is like comparing sudden death to death by a thousand paper cuts.

 

[BGinGA]

You all need to realize that both the question and the answer to that question were very specific.
....the question asked was very limited and precise and that's what the answers should address.

I think everybody who responded realizes that. But the answer to the question asked is pretty useless, as has been pointed out by several others.

I think Abby shed an excellent amount of illumination on the subject, and by only specifically answering the narrow scope of the question, we would not have had that gem posted.

 

[chkmte]

I'm just going to go on the record here and say that "I think" @abby99 is actually Chris Ferguson. "Lady Luck" my ass.

 

[Chicken Rob]

I'm just going to go on the record here and say that "I think" @abby99 is actually Chris Ferguson. "Lady Luck" my ass.

I've met her in person. She makes a better impression than Ferguson's public persona.

 

[mummel]

Abby what do you do for a living. You seem wicked smart.

 

[OklahomaBill]

28.50

 

[AWenger]

(Note: Implied odds are not part of the discussion)

Implied odds do exist in real life for a poker hand that's just on the flop.

I would draw every time to an open ended straight if my opponent only bet $25 into a pot of $100.

I do like your idea of a rule of thumb to consider bet sizing based on estimated opponent draws. Needs some tweaking though, IMO.

 

[abby99]

Abby what do you do for a living. You seem wicked smart.

I've worked as an accountant, an asset/liability manager, and a business/grants administrator. Yeah, I sense a theme there.

Poker is applied math, right? (y) :thumbsup:

 

[chkmte]

Implied odds do exist in real life for a poker hand that's just on the flop.

I would draw every time to an open ended straight if my opponent only bet $25 into a pot of $100.

I do like your idea of a rule of thumb to consider bet sizing based on estimated opponent draws. Needs some tweaking though, IMO.

Agreed. However, I think trying to fine-tune an algorigthm would be nearly impossible based on the sheer number of situations and types of opponents.

 

[Chicken Rob]

Agreed. However, I think trying to fine-tune an algorigthm would be nearly impossible based on the sheer number of situations and types of opponents.

Are you trying to build an algorithm to use in live games, or are you building a bot?

I once built a bot that could beat small stakes. You'd be surprised how easy it is to reduce the seemingly infinite situations to a well defined set of situations.

You're better off just betting 2/3 pot every time you bet than trying to change your bet size based on opponents perceived hand until you become a world class hand reader, and then you won't need an algorithm.

 

[chkmte]

Are you trying to build an algorithm to use in live games, or are you building a bot?

I once built a bot that could beat small stakes. You'd be surprised how easy it is to reduce the seemingly infinite situations to a well defined set of situations.

You're better off just betting 2/3 pot every time you bet than trying to change your bet size based on opponents perceived hand until you become a world class hand reader, and then you won't need an algorithm.

No, not trying to build an algorithm. I just love numbers, love learning, love creating. I find the mathematics of the game interesting. I'll say this, prior to this discussion - I was a good poker player but I'm not sure I could back up many of my bets with solid reasoning. Bottom line, I'm just trying to shore up my game and get a better understanding of the numbers. Little shortcuts (IMO) allow me to remember just a little bit more.

Tim

 

[chkmte]

On the flip-side @Chicken Rob , keeping this little nugget in mind would allow me to make better decisions as a potential caller as well...

 

[Chicken Rob]

On the flip-side @Chicken Rob , keeping this little nugget in mind would allow me to make better decisions as a potential caller as well...

Rule of 2 and rule of 4 get you to pretty good estimates for making calls when you think you are behind. That is for sure.

It's not clear where you are in your poker math learning curve, but rule of 2, rule of 4, and 2/3 pot will take you pretty far as you gain experience. Adjusting based on hand reading is probably more important than deeper math at that point.

The best way to really improve is get a low stakes game with skilled players together. It does amazing things for your game.

 

[chkmte]

We have low-stakes here. We have skilled players here. We do not have low-stakes, skilled players here. That's not really an exaggeration either. It's pretty sad.

 

[Chicken Rob]

We have low-stakes here. We have skilled players here. We do not have low-stakes, skilled players here. That's not really an exaggeration either. It's pretty sad.

You need to start building a low stakes game with skilled players on your own. It needs to be people who want to play together and win, but aren't just in it for the money.

Our local PCF game is a bunch of people from a 75 minute driving radius who all beat our regular home games and casino games, and we get together and play .25/.50. This .25/.50 game is probably the toughest game most of us ever get to play in. The soft players don't play in them that often. The tough players just keep getting better.

@guinness, @bergs, @GrumpyCatMalaka, @Jimulacrum, @iBetOnEverything, @ssanel54, @JoseRijo, are all regulars in the game, @snooptodd, @jbutler, @gopherblue, @manamongkids, @matill are all semi-regulars. My apologies to any regs I've overlooked.

Find a local PCF meetup group, and you'll find people serious about the game who are generally willing to play low stakes.

The Chicago crew with @Jeff and @abby99 who both host, @Ben has a game, @courage and @BGinGA and @H|Q allhave games. @k9dr and @Anthony Martino have communities.

The Maryland PCFers may still have a game, @MikesDad seems to be keeping it going now that @krafticus is on hiatus from hosting.

There are more here as well, these are all just the people I've played with.

Look to see if you're near a community like these, and be a good guest. If there isn't one near you, you gotta start building one.

When I first moved to NH, I built a home game. They were mostly very inexperienced. 6 years later, we still play and they're all much better. one won a meetup main event tournament, and another is a former SOHE world champion.

I don't mean to say stop learning the math. Just want to broaden your perspective about great ways to get better.

 

[DrStrange]

The value of basic math also declines as stack depth increases. You don't need much more than math and patience to win playing 50bb stacks. Playing 500bb stacks is essentially a different game altogether.

 

[Psypher1000]

It needs to be people who want to play together and win, but aren't just in it for the money.

Sounds like The Armory to me! Or that's the intent, at least.

"Ferrum Ferro Acuitur"

 

[Chicken Rob]

Sounds like The Armory to me! Or that's the intent, at least.

"Ferrum Ferro Acuitur"

Exactly!

 

[Chicken Rob]

The value of basic math also declines as stack depth increases. You don't need much more than math and patience to win playing 50bb stacks. Playing 500bb stacks is essentially a different game altogether.

This is another reason our .25/.50 game is soawesome. We start 250bb deep, and allow rebuys to half the biggest stack. We are often playing closer to >1000 bb effective late night. Shit gets real at that point. And learning to play deep is awesome.

 

[Jimulacrum]

This is another reason our .25/.50 game is soawesome. We start 250bb deep, and allow rebuys to half the biggest stack. We are often playing closer to >1000 bb effective late night. Shit gets real at that point. And learning to play deep is awesome.

You've left out the major detail that we barely play hold'em at all. Hold'em is like a novelty game. It's all about the sick Omaha variants that threaten to produce multi-way all-ins 2 or 3 times per orbit.

I'd love to see someone try to math his way to the right decision at a hand of SOHE. The game starts with 75 friggin' cents in the pot. Before you know it, 5 guys are going to a 400 BB flop with 800 BB effective behind. Flop gives you the second-nut flush in Omaha and TPTK in hold'em, and it goes pot-shove in front of you. Go ahead, try to range someone, I dare you.

 

[Psypher1000]

Go ahead, try to range someone, I dare you.

Nuthin' but air

 

[Chicken Rob]

Nuthin' but air

I don't think we've met yet, but it sounds like you know our player pool.

 

[Chicken Rob]

You've left out the major detail that we barely play hold'em at all. Hold'em is like a novelty game. It's all about the sick Omaha variants that threaten to produce multi-way all-ins 2 or 3 times per orbit.

I'd love to see someone try to math his way to the right decision at a hand of SOHE. The game starts with 75 friggin' cents in the pot. Before you know it, 5 guys are going to a 400 BB flop with 800 BB effective behind. Flop gives you the second-nut flush in Omaha and TPTK in hold'em, and it goes pot-shove in front of you. Go ahead, try to range someone, I dare you.

But before we got to the circus, we started with just Holdem. Later we added PLO. Then the dam broke.

We even went through a phase of playing "learnin' games" where any player could get a stray discussion at the conclusion of any hand. This did a lot for many of us.

 

[Adam Crowley]

This math has some good applications in shallow stack tournament play, it will give you a decent foundation of just how small you can cbet flops and turns and still force a mistake from opponents. Implied odds are less important in this situation as stack depths are shallower and hand reading is generally a lot easier.

 

[RainmanTrail]

Wait, people use math at the poker tables?