[FF] “Stan, what are the odds of flopping a royal flush?”, Figaro the Fish wondered.

[SS] “I don’t know it off the top of my head, but I have a calculator app, and we can work it out”, replied Stan the Stat. “Start with the number of five-card poker hands. There are 52 cards in the deck, and you want to select 5 of them. That’s 52-choose-5, which is called the Binomial Coefficient^{1} in Stats 101 and is 52 factorial divided by 5 factorial divided by (52 – 5) factorial… That’s 2,598,960 possible hands. Four of those are royal flushes, so your chance is 1 in 649,740 or 649,739:1.”

[YY] Yuri the Young Gun interjected, “Wow, I’ve played almost two million hands online and only flopped a royal flush once, so I guess I’m a little unlucky.”

[SS] “Only a tiny bit. You didn’t see the flop maybe half of time you had suited honors because you either took it down preflop with a raise or folded (are you really playing King-Ten suited under the gun?).”

[FF] “What about the odds of flopping less than a pair?”

[SS] “That’s a little harder, but using my favorite “choose” formula a couple times, we can work it out. We need to count the number of hands that aren’t flushes, aren’t straights, and don’t have any pairs or better. There are 4^5 ways for the suits to be arranged^{2}, of which 4 are flushes, so we have a factor of 4^5 – 4 = 1,020 for the suits. There are 13-choose-5 ways of selecting 5 different denominations, but 10 of those are straights (A-high down to 5-high), so we have a factor of 13-choose-5 – 10 = 1,277 for the denominations. Multiplying gets you 1,312,740, which is 50.12% of 2,598,960.”

[FF] “But that’s over half! I thought you hit the flop only around a third of the time.”

[SS] “You’re forgetting that some of those times you have a pair only because it’s on the board. The odds of a paired flop are right around 1-in-6^{3}, and one half minus one sixth is one third.”

#### Footnotes:

- It’s also called a combination without repetition.
- In statistics, it’s called a permutation with repetition. Order matters in a permutation but not in a combination. It may be weird to think that order matters when we’re talking about a poker hand, but it does here because we’re enumerating all the possibilities, not talking about a specific hand (which would show up multiple times in different orders).
- The number of ways to have one pair is 13 denominations times 4-choose-2 equals 6 ways to pick the two suits, times any third card of a different denomination, which is of course 48 (52 – 4). 13 times 6 times 48 is 3,744. Divide that by the total number flops, which is 52-choose-3, or 22,100, and you get 16.94%. The odds of a three of a kind flop can almost be ignored at 0.24% (13 denominations times four ways of missing one suit = 52 / 22,100).

#### Related Links:

- For the remaining hands, see the Wikipedia article on 5-card poker hands (note that the explanation for “No pair” incorrectly subtracts from the total number of hands).