Some of the techniques of combinatorics, or the study of counting, can be applied to calculate the probabilities of drawing certain types of hands in poker. The probability of being dealt a flush is relatively simple to find, but is more complicated than calculating the probability of being dealt a royal flush.
We will not be concerned with the order in which these cards are drawn, so each hand is a combination of five cards taken from a deck of 52 cards. There are a total number of C(52, 5) = 2,598,960 possible distinct hands. This set of hands forms our sample space.
Straight Flush Probability
We start by finding the probability of a straight flush. A straight flush is a hand with all five cards in sequential order, all of which are of the same suit. In order to correctly calculate the probability of a straight flush, there are a few stipulations that we must make.
We do not count a royal flush as a straight flush. So the highest ranking straight flush consists of a nine, ten, jack, queen and king of the same suit. Since an ace can count a low or high card, the lowest ranking straight flush is an ace, two, three, four and five of the same suit. Straights cannot loop through the ace, so queen, king, ace, two and three are not counted as a straight.
These conditions mean that there are nine straight flushes of a given suit. Since there are four different suits, this makes 4 x 9 = 36 total straight flushes. Therefore the probability of a straight flush is 36/2,598,960 = 0.0014%. This is approximately equivalent to 1/72193. So in the long run, we would expect to see this hand one time out of every 72,193 hands.
A flush consists of five cards which are all of the same suit. We must remember that there are four 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.
Rankings and Probabilities
We can see from the above that the ranking of each hand corresponds to its probability. The more likely that a hand is, the lower it is in ranking. The more improbable that a hand is, the higher its ranking.