Wikipedia:Reference desk/Archives/Mathematics/2013 January 4

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January 4

Starting hand matchups in Texas holdem

Since you get dealt 2 cards, there are C(52,2) = 1326 starting hand combinations for one player in Texas hold'em. However, there are only 169 distinct combinations, since some of those 1326 (such as A♦ K♠ and A♣ K♥) are equally likely to form any specific hand on showdown. Now, I'm wondering how many distinct combinations are there for two starting hands? N.B. sometimes you can switch suits and get the same distinct combination (such as 8♥ 8♣ vs 8♠ 8♦ = 8♣ 8♦ vs 8♥ 8♠ = 8♣ 8♠ vs 8♥ 8♦), but sometimes you get different combinations. For example, A♦ K♥ vs Q♦ J♥ ≠ A♦ K♥ vs Q♠ J♣, because QJ in the latter hand has more flush possibilites to outdraw the AK. I looked on the page Poker probability (Texas hold 'em), but the number there (690k) is far too big and, I presume, of similar nature as number 1326 in context of combinations of one starting hand. Any help is welcome, thanks in advance! 93.139.31.169 (talk) 13:32, 4 January 2013 (UTC)

I've given this some thought and it *really* gets ugly. For example A♦ K♥ vs a Queen and Jack off suited have 7 different possibilities: Q♠ J♥, Q♠ J♦, Q♠ J♣, Q♥ J♠,Q♥ J♦,Q♦ J♠ (and a for all but the 5th trade clubs and spades). What gets even uglier is things like A♠ 10♥ vs an off suited 9 and 8 where what suit the 9 & 8 are affects the probability of a Heart Straight Flush, but not a Spade Straight Flush.Naraht (talk) 17:20, 4 January 2013 (UTC)

This would probably require a computer simulation to solve. The most brute force method would be to try every possible combo of 4 cards (2 for each hand), which gives us 52!/48! possibilities, or 6,497,400. For each of those, you would then determine if they are distinct. You could check for each of the 169 distinct combinations for each hand, and that should limit you to 169², or 28561, distinct combos in both hands, at most. StuRat (talk) 19:47, 4 January 2013 (UTC)

Nope wouldn't be limited to 28561. As he pointed out in his original posting, there may be multiple different type of interactions between hands that by themselves wouldn't be viewed as different. For example, one of the 169 hands is "Ace and King off suited" which actually contains 12 different choices. One of the other 169 hands is "Queen and Jack off suited". *But* the relationship between the starting hands is different if the are A♦ K♥ vs Q♦ J♥ *or* if they are A♦ K♥ vs Q♠ J♣ because the chances of the Queen Jack off suited getting 4 cards of one of the suits to beat the Ace King with a Flush are better if the Queen Jack are in different suits than the Ace King. To put it another way, In the second combination of cards the 5,4,3 & 2 of spades (or any other 4 spades) faceup helps the Queen Jack hand, no equivalent set of four in a suit helps the Queen Jack in the first case.Naraht (talk) 21:23, 4 January 2013 (UTC)

Thanks for the input, guys. I think I've reached a solution. In poker, card ranks are ranked, but suits cannot be ranked, as all suits are equal in strength. Therefore the process of differentiating between suit formations in a hand is similar to differentiating between partitions of a set. For example, if a combination has 4 different ranks, there are C(13,4) ways to pick them and three ways to order them in groups (for example, for ranks 5 4 3 2, its 54 vs 32, 53 vs 42, 52 vs 43; for hands A and B, I consider A vs B and B vs A the same combination). Furthermore, there are B₄=15 ways to pick suits. I did similar for other combinations (such as pair vs two different cards, etc.), and got 49990 combinations. I'd appreciate it if anyone could confirm this number. 93.139.20.89 (talk) 23:16, 5 January 2013 (UTC)

Poker probability (Texas hold 'em)#Starting hands heads up gives 207,025. Bubba73 ^{You talkin' to me?} 03:10, 7 January 2013 (UTC)

Whoops, the note reduces it to 47,008. Bubba73 ^{You talkin' to me?} 03:12, 7 January 2013 (UTC)