Talk:Poker probability

Computing combinations

For posterity, below is a Python program that calculates the number of each kind of poker hand. It is roughly the same logic used in the article. (This code was written independently of the article.) --P3d0 07:42, Apr 13, 2005 (UTC)

   def choose( n, r ):
       if r > 0:
           return n * choose( n-1, r-1 ) / r
       else:
           return 1
   
   rf    = 4
       # 4 suits, all cards determined
   sf    = 4 * 9
       # 4 suits * 9 high cards (5 to K); all others determined
   four  = 13 * 48
       # 13 values for the quadruple, and any of the other 48 as a kicker
   fh    = choose(4,3)*13 * choose(4,2)*12
       # Any three of any value, and any two of a different value
   fl    = choose(13,5)*4 -sf-rf
       # Any 5 from any of the 4 suits, but not a straight flush
   st    = 4*10 * 4**4 -sf-rf
       # 4 suits * 10 high cards (5 to A), and the other 4 can be any suit, but exclude straight flush
   three = choose(4,3)*13 * choose(12,2)*4**2
       # Any three of any value, and any two of different values of any suit
   twop  = choose(4,2)*13 * choose(4,2)*12 * 44 / 2
       # Any two of any two values, and any kicker of a different value
       # Divide by two, otherwise each hand will be counted twice (eg. 8866K and 6688K)
   pair  = choose(4,2)*13 * choose(12,3) * 4**3
       # Any two of any value, and any three of different values, of any suit
   dud   = choose(13,5) * 4**5 -fl-st-sf-rf
       # Any five different values, any suits, except flushes or straights
   
   total = rf+sf+four+fh+fl+st+three+twop+pair+dud
   
   print "    Royal flush:", rf
   print " Straight flush:", sf
   print " Four of a kind:", four
   print "     Full house:", fh
   print "          Flush:", fl
   print "       Straight:", st
   print "Three of a kind:", three
   print "       Two pair:", twop
   print "           Pair:", pair
   print "        No pair:", dud
   print "          Total: %d (should be %d)" % ( total, choose(52,5) )

Output is this:

       Royal flush: 4
    Straight flush: 36
    Four of a kind: 624
        Full house: 3744
             Flush: 5108
          Straight: 10200
   Three of a kind: 54912
          Two pair: 123552
              Pair: 1098240
           No pair: 1302540
             Total: 2598960 (should be 2598960)

Royal flush as a separate hand

I'm removing royal flush. The problem is not that it's not a valid hand ranking...the problem is that it's a redundant hand ranking. The way the table is currently, a royal flush is not considered a straight flush, and this is just false. A royal flush is a particular type of straight flush. This makes the probability for straight flush incorrect. Revolver 03:18, 24 May 2005 (UTC)

Again, just to make it clear -- a royal flush is considered to be a straight flush. It is not a "separate category of hand ranking". Thus, removing it from straight flush category makes the numbers wrong. Revolver 03:25, 24 May 2005 (UTC)

Do you have a reference for this? Our own article on Hand (poker) lists the royal flush as a separate rank, as does the 1911 Britannica. I do understand it is redundant; in fact, I added this text to the poker hand article:

Interestingly, the royal flush is the only rank that has no effect on relative hand values. That is, if there were no such thing as a "royal flush", then every royal flush would still qualify as a straight flush, and would still outrank every other hand in poker (including the other straight flushes).

My own sense of elegance tells me that the royal flush should not be considered a separate rank, but what matters here is common usage. So, at the risk of starting a revert-war, I'm going to reinstate my changes; if you can find a reputable reference that indicates that the royal flush is not considered a separate rank, then I'll be happy to help visit all our poker-related articles and make them consistent on this point. --P3d0 03:54, May 24, 2005 (UTC)

Well, it's common knowledge in the industry (I am a poker dealer), but if you insist on a reference, here are some:

• Fundamentals of Poker, by Mason Malmuth and Lynne Loomis, p. 6.
• Official Dictionary of Poker, by Michael Wiesenberg, p. 200 and 236

Each of these authors are considered experts on poker and poker terminology. Revolver 04:42, 24 May 2005 (UTC)

The point is that the hand ranking classification should be a partition of all possible hands. You can't say, "well, it is a straight flush, but it's also a royal flush". You have to choose. The name is certainly used, but if a royal is to be considered a straight flush (as it is), it must be included so as to preserve the partition aspect of rankings. Revolver 04:46, 24 May 2005 (UTC)

Here's a good quote from Andrew Glaser, a poker player/writer:

"9) Straight flush. A very rare hand, unless many wild cards are being employed. Five cards in sequence that are also the same suit, for example, 5-6-7-8-9 of diamonds, would be a straight flush, and it beats four of a kind or any lower-ranked hand. Just as with regular flushes, suits are irrelevant: 5-6-7-8-9 of diamonds would split the pot with 5-6-7-8-9 of spades. Some people like to say that a Royal Flush (10-J-Q-K-A, all of the same suit) is a better hand than a straight flush, but technically, a Royal Flush is just the best possible straight flush. Don't spend too much time worrying about this distinction. Unless you're being cheated (like in the scene in the movie Honeymoon in Vegas), you will probably go your entire life without seeing a Royal Flush beat a straight flush."

Or, I should say, unless you work as a poker dealer! :) Revolver 05:46, 24 May 2005 (UTC)

Here's a somewhat obscure reason for not counting royal as a separate hand: it destroys the most popular ranking when bugs or jokers are used. In other words, if a bug/joker is used, 5 of a kind is usually ranked above any straight flush (including a royal flush), but if royal is a separate hand, mathematically it should rank ABOVE 5 of a kind. Revolver 05:49, 24 May 2005 (UTC)

Good point. I agree with that. Bubba73 (talk), 06:38, 30 November 2005 (UTC)

Ok, I'm convinced. Between the references you provided, and the 5-of-a-kind argument, I think that's enough justification to visit all the poker-related pages and make sure they consistently treat the royal flush as a nickname for a certain kind of straight flush, rather than a separate rank. (Incidentally, your "partition" argument is unconvincing because it would be equally valid (ie. not valid) if you use it, as written, to claim that a full house is not a separate hand from two pair.) --P3d0 13:28, May 24, 2005 (UTC)

The partition argument is convincing if you accept a royal flush is a straight flush. You have 2 choices: say that a royal flush is not a straight flush, and list them as 4 and 36 frequencies, or say it is, and list it just as 40. What you can't do is say a royal IS a straight, but then not count it in the table (and I was assuming you do consider it a straight flush). The difference with a boat is that a boat is not 2 pair. Revolver 10:33, 25 May 2005 (UTC)

Sorry, still not convincing. Arguably, a boat is indeed two pair, and the only reason a boat is not considered two pair is that every hand is given the highest rank for which it qualifies. In fact, this point of view is exactly the one expressed in our article as currently written. --P3d0 13:32, May 25, 2005 (UTC)

Well, that's the incorrect way of thinking about it. A boat is NOT two pair. Hands don't fall into multiple categories, and then you "choose the highest one". They only fall into a single category. It seems like pedantics, but the issue becomes more clear when you find yourself involved in the paradoxes of trying to order the hands when using wild cards or bugs. (In this case, the naive approach you take leads to logical contradictions.) You're defining rank in terms of frequency (rank = lower frequency), but also frequency in terms of rank (you only count the "highest rank" in the frequency count), which is circular. Here is how Malmuth defines the hand rankings:

1. Straight flush : Five cards of the same suit in sequence. An ace-high straight flush is called (nick-named) a royal flush (but is not a separate category).
2. Four of a kind : Four cards of the same rank, plus a fifth unrelated card. (Interestly, Malmuth incorrectly then states "the fifth unrelated card has no bearing on the hand's value", and this is false in community card games like holdem.)
3. Full house : Three cards of one rank and two cards of another rank.
4. Flush : Any five cards of the same suit not in sequence. (my emphasis)
5. Straight : Any five cards in sequence, not of the same suit (my emphasis)
6. Three of a kind : Three cards of the same rank, plus two unrelated cards (i.e. two cards of two other distinct ranks)
7. Two pair : Two cards of one rank, two cards of another, and one unrelated (i.e. distinct from the first two ranks) card.
8. One pair : Two cards of one rank, plus 3 unrelated cards.
9. No pair : Five unrelated cards (i.e. no pair, no straight, no flush)

This is also how Weisenberg defines hand rankings (e.g. he calls a straight "5 cards in sequence of mixed suits"). As it turns out, the naive approach just happens to lead to no contradiction in this particular case (that's one of the reasons it's useful to use 4 suits and 13 ranks...most other combinations produce contradictions in the rankings.) But, I think it's better to do like Malmuth and state precisely what "two pair" is, and is not. Another thing about the "choose the highest one" is that the royal flush option is kind of arbitrary. It just happens to be the nuts anyway. But, really ANY hand with frequency 4 could be chosen, e.g. a K-high straight flush. Then we could define "straight flush" to mean any straight flush (including royal) other than K-high, and define "K-high straight flush" as the best hand, since it only occurs 4 times, compared to 36 for straight. In fact, what's to stop someone from declaring Ks7c5c3c2s as "the nuts" because it is so rare (frequency 12, if you define it as K7532 with 753 same suit and K2 same suit). Maybe this is overkill to make a point. Revolver 00:43, 26 May 2005 (UTC)

This is good stuff. If you have references for all this (eg. I'm not familiar with Malmuth) it sounds like we should revamp the poker pages. --P3d0 19:16, May 26, 2005 (UTC)

This discussion seems to be stale, but the page is currently incorrect and inconsistent. The probabilities in the table are wrong (you don't need a calculator to see that - the probability for a royal flush is listed as 1/10th the probability of a straight flush, but the ratio of frequencies listed is 4/36). The note below the table incorrectly claims the listed straight flush includes royal flush. I'm doing what I see as the minimum to remove the inconsistencies and errors, which means I'm removing Royal Flush from the table, putting Straight Flush frequency to 40 and changing the oddds from 64,973:1 to 64,974:1 ( a typo I presume, the correct value of 2,598,960/40 is 64,974 ). The probability already there is correct for straight flush including royal. As far as I can tell this makes the table and page as a whole correct and internally consistent. If someone wants to add Royal Flush back in as a separate line they must recalculate the table, remove/update the note below the table and update the derrivations section to keep the page consistent and correct. --Mollymoo 16:32, 28 June 2006 (UTC)

"we"

This is a common mathematics style convention. This is just as much a math article as a poker article, so I see no reason to change it. Revolver 03:25, 24 May 2005 (UTC)

It is just as much an encyclopedia article as it is a math article. Since math articles are equally valid with and without "we", that is a moot point, and I think we should fall back to the more formal encyclopedic style. I actually don't feel all that strongly about it; the "we"s only disappeared again because of my revert for the royal flush issue (discussed above), but if you think they "we"s make the explanations clearer or more accessible, please feel free to reinstate them. --P3d0 03:58, May 24, 2005 (UTC)

Math articles may be "equally valid" with/without "we", but eliminating "we" just runs against mathematical style. If "formal encyclopedic style" trumps mathematical style, then all uses of "we" should be eliminated in math articles. Let me rephrase it this way: in mathematics, the use of "we" is formal style! Look up any number of proofs here. It's origins and purpose I believe are to make a tone of voice that's including the reader, rather than talking down to them. (I.e., "do this", "do that", is like a drill seargeant, whereas "we" is an invitation to discover something together.) Revolver 04:42, 24 May 2005 (UTC)

Ok, I agree it's better with "we" than without, but I still feel like we can do better. I have to go to work now, but I'll see if I can think of another way to describe the math that isn't so "procedural". --P3d0 13:31, May 24, 2005 (UTC)

It's not "procedural". It's just normal-sounding (to math people). But, it's really no big deal. Revolver 10:33, 25 May 2005 (UTC)

Wow, you don't want to give an inch, do you?  :-) It's "procedural" in the sense that it prescribes a sequence of actions that one performs to arrive at the correct result. The alternative is simply to describe the result itself. For instance, don't say "first we choose one of the four suits, then choose any two of the thirteen cards of that suit"; instead, say "The pair comprises any two of the thirteen cards of any of the four suits". --P3d0 19:20, May 26, 2005 (UTC)

I disagree, I think the "procedural" wording is better. This is how most elementary combinatorics arguments are presented...it's "imagined" that we are actually "doing" something, i.e. choosing balls out of an urn, cards out of a deck, flipping a coin, etc., and so we record the number of ways a certain outcome can occur as we go through the sequence of actions which determines the event we're trying to enumerate. Sorry if I seem stubborn. I'm simply standing firm by what my ears tells me sounds right, and I'm just saying that the way I worded it originally IS the way that it's normally worded in probability and statistics books, and the way it's normally worded by math people when talking to each other. Revolver 22:10, 7 Jun 2005 (UTC)

Eeeek, I just read the new wording. I really mean no offense, but it does sound much more confusing now. The whole point of these enumeration/counting type arguments is that we are "doing" something, "performing" a certain task and asking "how many ways"? The way it's worded now, the task is already done, and the performing of the tasks is linguistically hidden. Again, I hope you don't take this personally, I'm just giving my own opinion. Maybe we should have some non-math or relatively non-math people give their opinion. I'm just saying that the way it's worded, now, I have to verbally "reorganise" the statement, so that the procedural explanation is clear to me. Revolver 22:16, 7 Jun 2005 (UTC)

No offence taken. I'm not delighted with the current wording either, though I do think it's somewhat better than the "we" version. Let's see if anyone else has an opinion....

Procedural wording, part deux

Does anyone other than myself and Revolver have an opinion on the wording of the hand combination calculations? Here is a diff that contains the two contenders. --P3d0 18:03, Jun 8, 2005 (UTC)

The Royal "we" is standard in mathematics. I wouldn't bother to eliminate it, but it has already been changed and looks OK as it is now (without "we"). Bubba73 (talk), 06:42, 30 November 2005 (UTC)

Correction, not the "Royal we" but pluralis auctoris (inclusion of readers or listeners). Bubba73 (talk), 00:31, 1 December 2005 (UTC)

Seven-card hands

How about adding a table for the best 5-card hand within 7 cards (like the popular Texas Hold'em and 7-Card Stud)? The percentages/odds are a little different. Notably, no-pair is less frequent than 1- or 2-pair. Here are my calculations for the number of each kind of hand:

• Straight Flush: 41,584
• 4-of-a-kind: 224,848
• Full House: 3,473,184
• Flush: 4,047,644
• Straight: 6,180,020
• 3-of-a-kind: 6,461,620
• 2-pair: 31,433,400
• 1-pair: 58,627,800
• no-pair: 23,294,460
• total: 133,784,560

Spiffy sperry 22:07, 26 September 2005 (UTC)

I think it would be a great addition. --Hector 14:50, 25 July 2006 (UTC)

1:1

Shouldn't total probability be 1 : 1? --Hector 14:49, 25 July 2006 (UTC)

That's a matter of terminology: "probability" is expressed as a single real number from 0 to 1, often expressed as a fraction, where the numerator is total choices. Thus, a one-out-of-four chance has a probability of 1/4, or 0.25 (or 25%). But "odds" are expressed as a ratio of positive to negative outcomes, so a one-out-of-four chance is expressed as odds of 3:1 (read "three to one against"). Thus, a probability of 1.0 (certainty) is 0:1 odds. 1:1 odds represents 0.50 probability ("even money"). Odds are useful because they reflect the fair payout on a bet: if you were to bet $1 on something with a 0.25 probability, the fair payoff would be $3. --LDC 15:52, 25 July 2006 (UTC)

Reverted edits by 169.226.156.69

To explain my reversion of three edits by 169.226.156.69:

1. [1]: in standard poker variants, it is actually more likely that ace is both high and low. Although you are correct that the odds are different when ace cannot be either high or low, the odds should reflect the most common usage. It would be acceptable to ALSO include the odds for when ace is only high or low.
2. [2]: The odds updated in the table should not be based on ace only being low. In addition, if the ace is only low, then the odds of a flush and no-pair (high card) hand would also have to be adjusted. (The odds of these hands increase by an offsetting amount for the decrease in straight flushes and straights.)
3. [3]: The table does not separate out the royal flush from the straight flush. A royal flush is a straight flush and is included in the odds for a straight flush.

—Doug Bell ^talk 22:24, 3 December 2006 (UTC)

Probability of a flush

What is my failure if I calculate the number of possibilities for a flush this way?:

${4 \choose 1}{13 \choose 5}{47 \choose 2}=5,564,988$

(If I take the 2 kickers of the 47 cards, I have also the 5, 6 and 7-card flush possibility)

87.181.53.244 11:29, 29 December 2006 (UTC)

The failure is that you are counting 6 and 7 card flushes multiple times. This happens because the same cards can be selected in both the ${\begin{matrix}{13 \choose 5}\end{matrix}}$ and ${\begin{matrix}{47 \choose 2}\end{matrix}}$ terms. For example, choosing ♥; 2, 4, 5, 9, J; 6♥, Q♣ for the terms and choosing ♥; 2, 4, 6, 9, J; 5♥, Q♣ both produce the same hand. When calculating combinations, you have to structure the equation such that the terms are exclusive. —Doug Bell ^talk 19:13, 20 January 2007 (UTC)

I should also have mentioned that you are also counting straight flushes, four-of-a-kind, and full houses in your equation. The higher ranked hands have to be eliminated. —Doug Bell ^talk 19:53, 20 January 2007 (UTC)

Probability of Full House

Brushing up my maths skills, could you explain what's wrong with the following reasoning to calculate combinations for a full house (in the special case of a 3 cards + 1 pair + 1 pair), as mentioned in the 7 cards chapter.

$13{4 \choose 3}x12{4 \choose 2}x11{4 \choose 2}=247,104$

Which is by the way EXACTLY half of the calculate value, so I'm guessing I'm missing something. But what's wrong with the reasoning first you chose any 3 out of 4 of 13 options, then 2 out of 4 out of the remaining 12, then 2 out of 4 out of the remaining 11? —Preceding unsigned comment added by 94.224.70.144 (talk) 21:00, 20 July 2009 (UTC)

Four of a kind

I notice that the computation for four of a kind is given as

${13 \choose 1}{48 \choose 1}=624$

which is the correct result. However is it not more accurate to say that there are 13 possible rankings and from each rank 4 are chosen from each suit; making the computation:

${13 \choose 1}{4 \choose 4}{48 \choose 1}=624$

this is probably a pedant's delight but I make this point because this page is used by persons (such as me) with no real interst in poker but lots of interest in combinatorics.

AWR

Since I'd also thought that should be changed, I've changed it. —Doug Bell ^talk 19:15, 20 January 2007 (UTC)

Have it is the 4 x 13 type trhee a kind? Or the 3 of a kind have 13 four a kind? (READER)

Other Decks

It has been heard of for poker games to be played on non-standard decks. Would it be acceptable to add a section generalising the probabilities of the hands? —The preceding unsigned comment was added by 220.240.252.135 (talk) 20:59, 3 May 2007 (UTC).

Expressing Odds

Remember that when expressing odds to do the subtract-one thing: a 1-in-4 probability is 3-to-1 odds, not 4-to-1. Several times now people have "fixed" the straight flush odds in this article to 64974 instead of the correct number of 64973, since it is expressed as odds, not probability. Please stop "fixing" this! --LDC 06:15, 14 August 2007 (UTC)

Leak of Explanations

Some parts of the calcs explanations are missing like here: "A three of a kind must consist of 5 of the 13 ranks, but the 10 combinations that form straights must be subtracted." It doesn't show how the 10 was calculated. —Preceding unsigned comment added by 189.58.240.130 (talk) 20:35, 9 March 2009 (UTC)

Straight Flush

Listed as 72,193.33 : 1, should be 72,192.33 : 1? I haven't checked the rest of them. 22nd May 2009 —Preceding unsigned comment added by 82.39.228.91 (talk) 13:29, 22 May 2009 (UTC)