Calculating the Probability of a Royal Flush in Texas Holdem with 9 Players

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SUMMARY

The probability of achieving a royal flush in Texas Hold'em with 9 players, where the objective is to produce a royal flush on the table, is calculated using combinatorial methods. The specific calculation for the board showing a royal flush, without considering the hole cards, is determined to be 1 in 649,740. This is derived from the formula: (20/52) * (4/51) * (3/50) * (2/49) * (1/48). Understanding these calculations is essential for grasping the complexities of poker probabilities.

PREREQUISITES
  • Understanding of Texas Hold'em rules and gameplay
  • Basic knowledge of probability and combinatorial mathematics
  • Familiarity with card combinations and their significance in poker
  • Ability to interpret mathematical formulas and fractions
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  • Learn about the impact of hole cards on poker hand probabilities
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Zythyr
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How would I calculate the probability of getting a royal flush in a game of Texas Holdem with 9 players sitting on a table playing not against each other (without betting), but with the goal of producing a royal flush on the table. It doesn't matter which player gets the royal flush. The goal of the whole table is accomplished when a royal flush is attained.
 
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I think you will have to consider a lot of different cases separately. Only one of them is really easy, so I'll just do that one. The probability that the board will be a royal when we have no information about the hole cards is

[tex]\frac{20}{52}\cdot\frac{4}{51}\cdot\frac{3}{50}\cdot\frac{2}{49}\cdot\frac{1}{48}=\frac{1}{649740}[/tex]
 

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