Saturday, January 23 2010: Hand Shapes
Can someone who knows how to do math, explain this concept of hand shapes to me? Why are there only 16 hand shapes out of the 16,432 distinct hands (regardless of suit).
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The 270,725 starting hands can be reduced for purposes of determining the probability of starting hands for Omaha—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether cards in the hand share the same suit. Of the 270,725 combinations, there are 16,432 distinct starting hands grouped into 16 shapes. Throughout this article, hand shape is indicated with the ranks denoted using uppercase letters and suits denoted using lower case letters. For example, the hand shape XaXbYaYc is any hand containing two pair (XX and YY) that share one suit (a), but not the other suits (b and c). The 16 hand shapes can be organized into the following five hand types based on the ranks of the cards.
Rank type Shapes Distinct hands Combos Probability Odds
XXXX: Four of a kind 1 13 13 0.0000480 20,824 : 1
XXXY: Three of a kind 2 312 2,496 0.00922 107 : 1
XXYY: Two pair 3 234 2,808 0.0104 95.4 : 1
XXYZ: One pair 5 5,148 82,368 0.304 2.29 : 1
XYZR: No pair 5 10,725 183,040 0.676 0.479 : 1
TOTAL 16 16,432 270,725
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The 270,725 starting hands can be reduced for purposes of determining the probability of starting hands for Omaha—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether cards in the hand share the same suit. Of the 270,725 combinations, there are 16,432 distinct starting hands grouped into 16 shapes. Throughout this article, hand shape is indicated with the ranks denoted using uppercase letters and suits denoted using lower case letters. For example, the hand shape XaXbYaYc is any hand containing two pair (XX and YY) that share one suit (a), but not the other suits (b and c). The 16 hand shapes can be organized into the following five hand types based on the ranks of the cards.
Rank type Shapes Distinct hands Combos Probability Odds
XXXX: Four of a kind 1 13 13 0.0000480 20,824 : 1
XXXY: Three of a kind 2 312 2,496 0.00922 107 : 1
XXYY: Two pair 3 234 2,808 0.0104 95.4 : 1
XXYZ: One pair 5 5,148 82,368 0.304 2.29 : 1
XYZR: No pair 5 10,725 183,040 0.676 0.479 : 1
TOTAL 16 16,432 270,725
Steve wrote:
There's only one way to have four of a kind: all cards match in rank, with all four suits. That's one shape.
If you have three of a kind, the fourth card can match one of the three by suit (one shape) or not match any of them by suit (another shape). So two shapes.
If you have two pair, both cards in the high pair can match both cards in the low pair by suit (one shape), or one card in the high pair can match one card in the low pair by suit (one shape), or the all the cards can have distinct suits (one shape). Three shapes.
If you have one pair, the two single cards can have the same suit, which matches one card in the pair (1). Or the two single cards can have the same suit, not matching any card in the pair (2). Or the two single cards can have different suits, each matching one of the cards in the pair (3). Or the two cards can have different suits, only one of which matches one of the cards in the pair (4). Or the two cards can have different suits, none matching the cards in the pair (5).
If you have no pair, all cards can have different suits (1). Or two cards can have the same suit (2). Or three cards can have the same suit (3). Or it's a flush (4). Or two cards match by one suit and the other two match in a different suit (5).