Saturday, January 17, 2009

Venn Diagram of Poker Hands: Solution

Venn Diagram of Poker Hands:


FOUR OF A KIND and FULL HOUSE are both inside THREE OF A KIND and TWO PAIRS, but do not overlap.

THREE OF A KIND and TWO PAIRS are both inside PAIR.

ROYAL FLUSH is inside STRAIGHT FLUSH.

STRAIGHT FLUSH is inside STRAIGHT and FLUSH.

PAIR, STRAIGHT and FLUSH are all inside HIGH CARD.

Because any hand has a HIGH CARD and HIGH CARD includes all the others, HIGH CARD is identical to ALL HANDS.

The above diagram is topologically correct, I think, so it doesn't really matter what sizes or shapes the boundaries take.


An additional problem, for those who love easy combinatorics, would be to show the numbers and probabilities for all the regions. This requires a slightly different interpretation of the labels: for example, HIGH CARD is interpreted to mean "HIGH CARD ONLY" which excludes PAIR, FLUSH and STRAIGHT.

5 comments:

emma kant said...

This is really quite wrong. For just one example, flush and pair are not disjoint as you represent them. You can have a pair in a flush. Although there is no poker hand called "flush with a pair", nevertheless, the sets of hands matching "one pair" and "flush" are not disjoint as you represent them to be.

Mike Huben said...

This is simple 5-card poker, not a variant such as Texas Hold-Em. Since a flush has all cards of the same suit (not color!), and each suit has only one of each rank, you cannot have a pair in a flush.

The easiest way to see this is to try to make a hand that has a flush and a pair.

emma kant said...

Oops, for some reason I was thinking here that a flush WAS five cards all of the same color, although I actually did know it was five cards all of one suit. Weird.

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