Big Two and n-Card Poker Probabilities

Between the poker hands of straight, flush, and full house, which hand is more common? In standard 5-card poker, the order from most common to least common is straight, flush, full house. The same order is true for 7-card poker such as Texas hold'em. However, is the same true for n-card poker for larger n? We study the probability of obtaining these various hands for n-card poker for various values of n≥5. In particular, we derive closed expressions for the probabilities of flush, straight and full house and show that the probability of a flush is less than a straight when n≤11, and is more than a straight when n>11. Similarly, we show that the probability of a full house is less than a straight when n≤19, and is more than a straight when n>19. This means that for games such as Big Two where the ordering of 13-card hands depends on the ordering in 5-card poker, the rank ordering does not follow the occurrence probability ordering, contrary to what intuition suggests.