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A flush is a hand that contains five cards all of the same suit, not all of the sequential rank, such as Q♣ 9♣ 7♣ 2♣ 3♣ (a king-high flush). It ranks below a full house and above a straight. Under ace-to-five low rules, flushes are not possible (so K♥ 8♥ 4♥ 3♥ 2♥ is a King-high hand).
Each flush is ranked first by the rank of its highest-ranking card, then by the rank of its second highest-ranking card, then by the rank of its third highest-ranking card, then by the rank of its fourth highest-ranking card, and finally by the rank of its lowest-ranking card.
Example: K♦ J♦ 9♦ 6♦ 4♦ ranks higher than Q♣ J♣ 7♣ 6♣ 5♣, which ranks higher than J♥ 10♥ 9♥ 4♥ 2♥, which ranks higher than J♠ 10♠ 8♠ 6♠ 3♠, which ranks higher than J♥ 10♥ 8♥ 4♥ 3♥, which ranks higher than J♣ 10♣ 8♣ 4♣ 2♣. Flush hands never differ by suit alone. Here 10♦ 8♦ 7♦ 6♦ 5♦ and 10♠ 8♠ 7♠ 6♠ 5♠, are considered as an equal rank.
A flush beats a straight. A flush is any five cards all of the same suit (i.e., all diamonds or all spades, etc.). If two or more players share a flush then the player with the highest card (all the way to the fifth card if necessary) in the flush wins.
A flush consists of five cards which are all of the same suit. We must remember that there are 4 suits each with a total of 13 cards. Thus a flush is a combination of five cards from a total of 13 of the same suit. This is done in C(13, 5) = 1287 ways. Since there are four different suits, there are a total of 4 x 1287 = 5148 flushes possible.
Some of these flushes have already been counted as higher ranked hands. We must subtract the number of straight flushes and Royal Flushes from 5148 in order to obtain flushes that are not of a higher rank. There are 36 straight flushes and 4 royal flushes. We must make sure not to double count these hands. This means that there are 5148 – 40 = 5108 flushes that are not of a higher rank.
We can now calculate the probability of a flush as 5108/2,598,960 = 0.1965%. This probability is approximately 1/509. So in the long run, one out of every 509 hands is a flush.
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