Talk:Poker probability (Texas hold 'em)

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You know a big problem with this page is I read "barring the miracle flush or straight" a lot. Why would we bar it? It's a part of the game it needs to be figured into the odds.....

Contents

[edit] Values in "odds" columns

Is it just me, or are all the numbers in the "odds" column 1 lower then what they should be?

I will change them but if someone else points out where my math is going wrong, then my apoligies.

I can assure you, it was correct before. It's okay, it's a common mistake. The formula is: the odds are defined by (1/p) − 1 : 1, where p is the probability. So, if p = 1/2, the odds should be 1:1, not 2:1. What you're forgetting about is that its the relative frequency of winning to losing, not winning to total action. Revolver 07:34, 5 Jul 2004 (UTC)

[edit] I'd like to see...

  • If you flop a flush, the probabilities that another opponent flopped a higher flush. I'm picturing a table high cards down the side and number of opponents across the top.--Toms2866 03:01, 10 May 2006 (UTC)
    OK, I have those...I'll add that, but it may be a little while as I'm going to be out of town. —Doug Bell talk•contrib 01:25, 11 May 2006 (UTC)
  • Also, if you flop a (say) heart flush, odds someone has a single bigger heart (drawing to a bigger flush) by number of opponents. Brian Alspach has some interesting stats about losing flushes (assuming all opponents see the river). [1] I was thinking about putting his findings in, but sans the derivations (too complicated for article). Think it would be a good addition?--Toms2866 02:25, 11 May 2006 (UTC)
  • What I miss a little bit is the probability to hit something on the flop. E.g. with JTs, how likely is it to have a straight draw (open-ended and/or gutshot), or a flush draw, a pair, two pair,... I found some useful tables at Mike Caros University of Poker Library (XVIII-XXVI for Hold 'em) with some numbers (e.g. 8.14:1 against having a flush draw). WhoCares01 21:43, 10 January 2007 (UTC)

[edit] Starting hands

I think the formula {52 \choose 2} = 1326 is going to be pretty incomprehensible to 99% of readers. Would it not be clearer to say 52 times 51 divided by 2 = 1326? We non-mathematicians can understand that the first card may be any of 52, that for each first card the second may be any of 51, and that we divide by 2 because each combination may be produced by either Card A followed by Card B or Card B followed by Card A.

In any case, for holdem, 169 is the magic figure. Distinguishing hands such as 5C 3H from 5C 3S is irrelevant and misleading. Obviously those two examples play exactly the same and have the same chance of winning. I've changed this, but my derivation is pretty clunkily worded, and probably not necessary anyway. Stevage 17:23, 3 December 2005 (UTC)

I changed the discussion to include both {52 \choose 2} = 1,326 and 52 × 51 ÷ 2 = 1,326. I think by discussing both means of representation, the combinatorial math can be introduced in a way that makes is comprehensible to at least most of the other 99% of readers. It gets really messy trying to show the calculations without binomial coefficients once you get beyond choosing two from a set. I also expanded and hopefully made less clunky the explanation of the 169 different strength starting hands. – Doug Bell talk•contrib 07:23, 6 February 2006 (UTC)

What does the "any specific (no/)pair" phrase mean in the starting hands table? Does that mean that any pair in your starting hand as the same odds as AA? That doesn't seem to make sense. Revise the numbers please and link this to combinatorial game theory. 70.111.251.203 23:28, 11 February 2006 (UTC)

That wording existed in the article before I began editing it. The word specific is the key to understand the meaning. AA and KK have the same odds. AK and T2 have the same odds. AKs and 78s have the same odds. Each of these is an example of a specific hand with the same characteristics (hand shape). – Doug Bell talk•contrib 11:47, 23 February 2006 (UTC)

[edit] Calculations for probability of facing larger pocket pairs from multiple opponents

The equations I entered are wrong. I will fix them soon, but please leave them there for the moment unless you want to fix them. They are actually reasonably close approximations. The function cannot use (1 - psingle)players as the events are not independent. The calculation and explanation needs to use (players × psingle) - pmultiple where players is the number of opponents faced, psingle is the probability that a single opponent has a higher pair and pmultiple is the probability that multiple opponents have a higher pair. – Doug Bell talk•contrib 23:08, 6 February 2006 (UTC)

OK, I fixed the equation and the results table. – Doug Bell talk•contrib 01:28, 8 February 2006 (UTC)

[edit] Latest changes

Just dropping by to say great work on the latest changes. This article could become one of the best within the WikiProject when completed. Look forward to seeing it progress! Essexmutant 00:04, 10 February 2006 (UTC)

[edit] Combinatorial game theory and complexity

Added them to the related links, since they are part of it. 128.6.175.60 20:24, 20 February 2006 (UTC)

[edit] Pictures

Though I love what's going on with this article, there seems to be an overdose of pictures. I'm on a DSL connection and all the pictures don't load within a short time. Is there a way we can reduce the amount, while still keeping all the good information? Perhaps just a single picture file that has the whole chart, instead of a chart with a lot of pictures? 128.6.175.60 20:35, 20 February 2006 (UTC)

First, some of it depends on the general responsiveness of Wiki. The "pictures" are all the math equations. If you set your preferences under the "Math" tab to "HTML if possible or else PNG", many of the equations will be rendered as HTML instead of images. There will still be a lot of images, but probably less than half. Try this and let me know how it works for you. – Doug Bell talk•contrib 21:22, 20 February 2006 (UTC)

[edit] 3 mistakes so far (please check)

The 4th formula in the chapter "starting hands against multiple opponents" seems incorrect: "... and against n opponents is H =..." The passage is "50-2k" , it should be "52-2k" or it does not work out.

I think you are forgetting that 2 cards are already in the player's hand, leaving only 50 cards remaining in the deck to be distributed. – Doug Bell talk•contrib 18:42, 8 March 2006 (UTC)

Chapter "Pocket Pairs": The Link "Probabilites during play" does not work!

Fixed. I renamed the section and forgot to change the link, thanks for pointing that out. – Doug Bell talk•contrib 18:42, 8 March 2006 (UTC)

The chapter "Hands with one ace": The formula contains a "*2" in the second half ("3/50 * (13-x)*4*2/49"). Damn, where does this *2 come from? Without it, it should be in the end: 3/1225 + [6*(13-x)/1225].

Added an explanation for the *2. – Doug Bell talk•contrib 18:42, 8 March 2006 (UTC)

Correct me, if I am wrong. I'd be glad if (in case these are mistakes) get corrected soon.

Thanks.

Sam

Germany

Thank you for your comments, please feel free to either provide additional feedback or simply edit any problems you find in the article. – Doug Bell talk•contrib 18:42, 8 March 2006 (UTC)

[edit] Head-to-head probabilities for different starting hand matchups

I added a section on head-to-head matchup probabilities. It doesn't have the mathematical rigor of the other sections, but it may be the most useful section in the article from a practical "at the table" point of view. It certainly seems like something a reader might be looking for in this article. My personal opinion is that adding math rigor for this topic would consume an inordinate amount of space with little added value for 99% of readers. --Toms2866 00:07, 24 March 2006 (UTC)

Well this is certainly open for discussion, but my thinking on the matter is that if all you want is tables of odds, there are many places on the Web to get those. So my philosophy in developing the article was to link the math and the probabilities. After all, the name of the article is "Poker probability" not "Poker odds tables". So I haven't put anything in without a discussion of the math.
However, the math for complete head-to-head comparisons is not practical. These situations are pretty much only determined through brute force, so I'm fine with the section you added and appreciate the contribution. I will probably tweak it a bit, in particular I've kept all the odds in a X : 1 format so that they can be easily compared. —Doug Bell talk•contrib 02:25, 24 March 2006 (UTC)

[edit] References

Not related to the above particular calculations, but I noted with pleasure that the chapter "Flopping overcards when holding a pocket pair" matches similar calculations by Brian Alspach: Overcard Calculations. He has a number of other interesting poker calculations that may be worthy of inclusion in this article. Examples include probabilities of straight completion by starting hand, probabilities of making a losing flush by starting hand, board suit and rank distributions, etc. See Poker Calculations by Brian Alspach. --Toms2866 02:44, 23 March 2006 (UTC)

Thanks, that is good site—better than the other references in the article. I've been having trouble finding quality references. The calculation for overcards is one of the simpler calculations, but it's nice to have independent verification. —Doug Bell talk•contrib 02:54, 23 March 2006 (UTC)

[edit] Move

I suggest that we move this article to Texas hold 'em probability. It is a simpler name and therefore better in my opion. --Maitch 16:58, 1 May 2006 (UTC)

There is a poker probability article so this one should definitely stay what it is to be consistent. 2005 19:40, 1 May 2006 (UTC)

I'm aware of that article, but subarticles doesn't have to use parenthesis. I would actually say it is more normal to not do it. History of France is a subarticle of France and it doesn't have the name France (history). --Maitch 19:52, 1 May 2006 (UTC)

It is certainly much more common to do it the way it is, as there are more than dozen of examples in Category:Poker gameplay and terminology that are structured with (poker) parentheses. I don't see any reason to go non-standard here, although I certainly agree it reads better as Texas Hold 'em probability. If nobody cares about going a non-standard way in a few days, go ahead and move it if you want. 2005 20:02, 1 May 2006 (UTC)
I don't have a strong preference either way, but my slight preference is to leave it named as it is for the reasons 2005 states. —Doug Bell talk•contrib 22:24, 1 May 2006 (UTC)

Well, in my opion there is a difference between e.g. Aggression (poker) and Poker probability (Texas hold 'em). First of all is Aggression (poker) not a subarticle of Aggression. Secondly, all the other articles you call standard use "(poker)", which is different from "(Texas hold 'em)", so this article is really alone in that category. --Maitch 22:38, 1 May 2006 (UTC)

I'm inclined toward not renaming the article. I think the current naming scheme makes it clearer that this article is a variant-specific discussion of the topic more generally covered in Poker probability. That said, my opinion is not strongly held.--Toms2866 23:32, 1 May 2006 (UTC)

[edit] Texas hold'em hands

Texas hold 'em hands should be included into this article. That chart that they have along with the formula below it would be handy in the probability department. 70.111.244.69 14:49, 29 July 2006 (UTC)

[edit] Shoudn't "\times" ("\times") be replaced by "\cdot" ("\cdot")?

It is my opinion as a LaTeX-amateur that the symbol "\times" ("\times") be replaced by "\cdot" ("\cdot"). It would be mathematically more correct. The symbol "\times" appears nearly 30 times throughout the article, so I thought I'd ask before changing it! --NicApicella 20:40, 20 August 2006 (UTC)

The x symbol is probably more easily recognised by those without a maths background. That's not a "no", but something to consider. Stevage 20:46, 20 August 2006 (UTC)
That was my reasoning in using the \times symbol. —Doug Bell talk•contrib
I slightly prefer the "\cdot". Any reader with adequate mathematical background to understand the equations will be familiar with the \cdot symbol. The \times symbol can be a awkward in equations using the variable "x".--Toms2866 17:02, 18 October 2006 (UTC)
I slightly prefer the \cdot myself, but I really worked hard to keep the math in the article approachable for everyone—the \times was just one small component of that effort. And while I agree in principle with the "awkward with 'x'" argument, TeX does at least make the two rather easy to distinguish. I suppose one option would be to replace x with some other variable in the equations that use it. —Doug Bell talk•contrib 17:29, 18 October 2006 (UTC)

[edit] Miscalculations in section "Head-to-head starting hand matchups"

"Probability" doesn't equal "Odds for" in this section. - Jack's Revenge 23:59, 9 December 2006 (UTC)

Actually it does. The confusing part with this table (one of the few that I didn't add to the article) is that the odds displayed are odds for the event happening. This contrasts with the rest of the odds in the article which are the odds against the event happening. Odds for are the inverse of odds againsts (i.e. 3 : 1 odds against an event happening are 1 : 3 or ⅓ : 1 for the event). So the odds in that table are calculated by the formula p / (1-p) and are correct.
The reasoning (I presume) behind using odds for in this case is that the favorable outcome for the player is represented by the odds in the table. —Doug Bell talk 11:56, 10 December 2006 (UTC)
Sorry. - Jack's Revenge 20:33, 12 December 2006 (UTC)

[edit] Chance of two suited cards making a flush?

Somewhere in this article it would be nice to see the chance of:

  1. ...two suited cards meeting two more of the same suit on the flop
  2. ...meeting 3 more of the same suit on the flop
  3. ...either way, two suited cards completing to a flush by the river. Stevage 10:51, 16 February 2007 (UTC)
OK, I'll add that. Same with connectors making a straight draw, a pair making trips. May be a few days before I get to it. —Doug Bell talk 18:38, 16 February 2007 (UTC)

[edit] Flopping overcards when holding a pocket pair

I think we should also point out, that these probabilities for overcard flopping also have flops that make pocket pair to come Three of a kind. For instance, there is 0,5696 probability for overcard to flop when holding JJ pocket pair, but there is also 0,1176 probability for one more J to flop. So there is 0,5696*0,8824 =0,50 probability for overcard to flop and J not to flop--Teveten 14:21, 14 March 2007 (UTC)

I'm a bit confused: You compute the probability for an overcard to be in the flop assuming, for example, having 4 Aces in the deck and choosing 3 cards out of 50. But since it means no danger to me with my pocket pair to see an overcard in the flop under these assumptions, I wonder if it might be more interesting to assume that my opponent already had an Ace, which reduces the remaining Aces to 3 and the deck to 49 cards. --Stefan 20:36, 21 August 2007 (UTC) —The preceding unsigned comment was added by 89.61.250.45 (talk)

[edit] Discussion/derivation

This article is a bit too pedagogical for an encyclopaedia article, imho. This article should really be answering the question "What are the chances of X happening in a game of Texas hold 'em.", for various X. Discussion explaining how these figures are calculated is not very relevant, or could be moved down into a footnote, or an explanatory "Derivation" section at the bottom. In particular, what's the point of the "When calculating probabilities for a card game such as Texas Hold 'em, there are two basic approaches." paragraph in the intro? Sure, there are two different ways you can calculate them - but so what? The article is primarily about poker, not about maths. Stevage 07:06, 21 March 2007 (UTC)

Well, I disagree somewhat with your last statement—the article is about poker and math equally. Just because you're coming at it from the poker perspective doesn't make the math view of it any less valid or relevant. However, your argument is reasonable. If you want to move it to a note, that's fine—I won't move it back. As to using the <ref> tag, I wasn't planning on using those for the notes. I'm planning (or anyone else is free to do so) on making a pass at some point to add reference citations for parts of the discussion. Those would be different than the notes and would go in a separare section. —Doug Bell talk 07:29, 21 March 2007 (UTC)

[edit] Wrong Question

To add to Stevage's comments and draw generalizations from them and some of the others:

When playing poker, you're not only interested in the odds of getting a hand, you're interested in the odds of beating your opponents hand. Given the cards you know about (hole cards and community cards), how strong is your hand? It's a comparison of the relative strength of your hand that's needed. So while I think the article provides important information, it's addressing the wrong question.

And I disagree on the mathematical validity point. The article is about applied mathematics. --71.202.189.23 23:44, 12 June 2007 (UTC)

Well of course a complete poker strategy needs both, but that doesn't make the information here irrelevant. Let's say, for example, one is drawing to a flush when the board is already paired; one needs to evaluate the situation as a whole, including how the opponent has bet, your knowledge of him and any possible tells, to decide whether or not the flush will be good if it hits. But once you decide that it will be, you still need to know your odds of hitting it to make the final decision. We're not offering the information here as if that were all one needs to know to play poker well--it's just one of many things one needs to know. --LDC 17:21, 15 June 2007 (UTC)
That's what a gaming or applied-math guide is for. This is an encyclopedia. The tone is great for a guide but wrong for an encyclopedia. Canuckle 03:45, 23 June 2007 (UTC)

[edit] Wrong statement about the nuts

I've added "(except for someone having a straight flush)" to the statement "if the flop comes with three 2s, any hand holding the fourth 2 has the nuts."

Is it wrong though? At that point, the best hand it is possible to make with the availible community cards is quad 2's. So that hand is the nuts. Although it is possible to hit running cards to make better quads or a straight flush. It depends how you define "the nuts". Is "the nuts" the best hand possible at that point, or the best hand possible after the river is dealt. It is not possible to make a straight flush on the flop if it is 222, only a backdoor straight flush *draw*, so the revision is not correct regardless of your interpretation of the nuts. What do people think? Maybe a wording like: "if the flop comes with three 2s, any hand holding the fourth 2 has the nuts. There is however a small probability that a player may hit running cards to make a straight flush or better quads, which would then become the nuts."? 82.46.37.37 16:53, 3 August 2007 (UTC)

[edit] Intro

The methods given are not really mutually exclusive, nor are they exhaustive. As the later sections make clear, there are a number of methods depending on your needs.

You can use a pure stochastic (Monte Carlo) method by generating a random sample set and averaging the outcome. In some cases you can use combinatorics to figure out the number of ways of making a given hand. In some cases you need very little math because you have a dominated hand and just need to compare your outs.

The article is great, I just thought the intro did not really address the tools that are available to a mathemetician or programmer when calculating probabilities.

[edit] Conditional

If you are holded suited cards, does that increase the odds that other players are holding suited cards too (in the other three suits). By about how much? Oyster Jimmy (talk) 06:13, 22 December 2007 (UTC)