Royal Flush Odds Change With the Cards Available

The odds for a royal flush are 1 in 649,740 on a five-card deal. In Texas Hold’em, where a player can make a hand from seven available cards by the river, the chance is 1 in 30,940.

Those figures answer different questions. Video poker introduces another distinction: the initial deal has the five-card odds, but drawing from a partial royal offers a much shorter conditional chance.

Choose the cards available and enter the number of independent attempts to compare the chance of seeing a royal.

These calculations assume a standard, fairly shuffled 52-card deck with no wild cards. A royal flush consists of 10-J-Q-K-A in one suit. UCLA’s poker-combinations notes identify four royal flushes among 2,598,960 possible five-card hands (UCLA Department of Statistics).

Five-Card Royal Flush Odds Are 1 in 649,740

There are four possible royal flushes: one each in clubs, diamonds, hearts and spades. The number of unordered five-card hands is C(52,5) = 2,598,960.

The probability is therefore 4 divided by 2,598,960, which reduces to 1 divided by 649,740. That is 0.0001539%, or odds against of 649,739 to 1.

The probability applies to a complete five-card hand drawn randomly from a standard deck. It does not change according to the royal’s suit.

“1 in 649,740” is a long-run frequency, not a required waiting period. A royal could appear on the next deal, and a player could also go through far more than 649,740 deals without seeing one.

For example, across 100,000 independent five-card deals, the probability of at least one royal is 1 − (649,739 ÷ 649,740)^100,000, or approximately 14.3%. Repeated play raises the cumulative chance, but no number of prior misses makes a royal due on the next independent deal.

Texas Hold’em Royal Flush Odds Are 1 in 30,940 by the River

In Texas Hold’em, five community cards are dealt by the river. Each player combines those cards with two private cards to make the best available five-card poker hand (Bicycle’s published rules). That gives one player seven available cards.

For each suit, fix the five cards required for its royal and choose the remaining two cards from the other 47. The number of favorable seven-card sets is 4 × C(47,2) = 4,324.

The total number of seven-card sets is C(52,7) = 133,784,560. Dividing 4,324 by 133,784,560 gives 1 in 30,940, or 0.0032321%. The corresponding odds against are 30,939 to 1.

This is the chance that one player’s seven available cards contain a royal when no cards are known. It is not the probability that somebody at a multiplayer table will make one, and it does not account for folded hands or known cards. For the dealing sequence and the use of hole cards and community cards, see Texas Hold’em step by step.

A player does not need to use both hole cards. A royal may use two, one or none of them, depending on the board. The seven-card calculation includes all of those possibilities because it asks only whether the required five suited cards appear among the seven available cards.

Video Poker Draw Odds Depend on the Cards Held

A five-card video-poker initial deal has the same 1-in-649,740 royal probability as any random five-card deal. Once the first five cards are visible, conditional draw odds are more useful than the initial-deal figure.

Assume a standard one-draw game with no wild cards and that all held royal cards belong to the same suit.

Four Cards to a Royal Give a 1-in-47 Draw

With four royal cards held, exactly one of the 47 unseen cards completes the hand. The chance on the one-card draw is therefore 1 in 47, or 2.1277%. The odds against are 46 to 1.

This figure applies only when the four held cards are four different ranks from 10 through ace in the same suit. Four high cards spread across different suits are not four to a royal.

Three Cards to a Royal Give a 1-in-1,081 Draw

With three suited royal cards held, the two missing royal cards must both appear in the two-card draw. There are C(47,2) = 1,081 possible unordered two-card draws, and only one contains both required cards.

The chance is therefore 1 in 1,081, or 0.09251%. The odds against are 1,080 to 1.

Two Cards to a Royal Give a 1-in-16,215 Draw

With two suited royal cards held, all three missing royal cards must appear in the three-card draw. There are C(47,3) = 16,215 possible unordered draws, with one exact three-card combination completing the royal.

The chance is 1 in 16,215, or 0.006167%. The odds against are 16,214 to 1.

Per-Draw Odds Are Not the Same as Final-Hand Frequency

The video-poker draw figures are conditional probabilities. The 1-in-47 figure, for example, assumes the player has already received four cards to a royal and holds them before drawing. It is not the chance of finishing any randomly started video-poker hand with a royal.

In 9/6 Jacks or Better, the final royal probability under the strategy used for the return calculation is about 0.00002476, or roughly 1 in 40,391 (Wizard of Odds pay-table analysis). That final-hand frequency is higher than the dealt-only probability because some initial hands can improve to a royal on the draw.

The 1-in-40,391 figure is tied to that game and its strategy calculation. Different pay tables, strategy choices or games with wild cards do not inherit the same final royal frequency. The five-card and seven-card headline probabilities describe standard-deck card combinations rather than the return or house edge of a particular machine.