Thus, the total number of royal straight flushes is Royal straight flush - A royal straight flush is a subset of all straight flushes in which the ace is the highest card (ie 10-J-Q-K-A in any of the four suits).Thus, the total number of straight flushes is: Straight flush - Each straight flush is uniquely determined by its highest ranking card and these ranks go from 5 ( A-2-3-4-5) up to A ( 10-J-Q-K-A) in each of the 4 suits.See also: sample space and event (probability theory). Of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. Derivation of frequencies of 5-card poker hands
However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two-that difference could affect the relative value of each hand when there are more cards to come. The number of distinct poker hands is even smaller. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits.
The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair. When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1. The royal flush is a case of the straight flush. Straight (excluding royal flush and straight flush) Mathematical expression of absolute frequencyįlush (excluding royal flush and straight flush) The nCr function on most scientific calculators can be used to calculate hand frequencies entering nCr with 52 and 5, for example, yields as above. (The frequencies given are exact the probabilities and odds are approximate.) Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, five-card hands).
The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.