Are the odds of obtaining certain hands in Hold 'Em poker the same as the 5-card odds we're accustomed to seeing?
Take the royal flush as an example. Traditional statistics for 5-card poker tell us that out of the 2,598,960 (52 choose 5) possible hands, one can obtain a royal flush in only 4 ways, yielding odds of 649,739-to-1. I'm sure this figure is familiar to a lot of you.
I would assert that in Hold 'Em poker, there are actually 133,784,560 (52 choose 7) possible hands (of 7 cards, not just 5) that one could see, and that there are 4,324 ways to obtain a royal flush. The 4,324 figure is obtained by multiplying the 4 royal flushes by the number of ways to fill out the remaining 2 cards (47 choose 2, which equals 1,081). These figures yield odds of 30,939-to-1, which are exactly 21 times more likely than the "traditional" 5-cards odds.
Am I stating something that's already known? Does anyone else agree with this or see any flaws with this thinking? Does anyone care?
