What are the odds against ‘flopping’ a royal flush in Texas hold ‘em poker?
An ace-high straight flush, or royal flush, consisting of ace, king, queen, jack and ten, all of the same suit, is the most valuable and, consequently, the most sought after, hand in Texas hold ‘em poker. However, the royal flush is, indeed, a rare bird and occurs in Texas hold ‘em poker once in every 40,000 hands, on average.
Each hand in Texas hold ‘em poker consists of a combination of five cards, from the two, known as hole cards, dealt face down to each player at the start of the hand, and the five, known as community cards, dealt face up in the centre of the table. Regardless of any other factors, the overall odds of achieving a royal flush in a seven-card game of this type are 30,940/1.
However, the community cards are dealt in three stages, the first of which consists of a series of three cards, collectively known as the flop. Consequently, ‘flopping’ a royal flush requires a player to be dealt two suited cards, ten or better – the odds against which are 32.2/1, in the first place – followed by a flop consisting of three specific cards, so the odds of doing so are much higher than forming a royal flush on the fourth, or fifth, card.
Indeed, it is possible to form 2,598,960 hands from a deck of 52 cards, but a royal flush can only be formed in one of four ways – clubs, diamonds, hearts or spades – so the odds of ‘flopping’ a royal flush are 2,598,960/4 or 649,739/1. In other words, according to one estimate, your chances of doing so are only marginally better than winning an Olympic gold medal during your lifetime.