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Every video poker enthusiast loves the excitement of hitting a royal flush while playing. Recently, one of my friends told me he hit a royal flush without holding any cards and wondered how rare it was. He even sent me a picture as proof.

Congratulations on hitting the royal flush, my friend! I’m happy to help you calculate the odds.

By developing Video Poker Magician, I have built a solid foundation for the video poker game calculation engine. With a few small tweaks, I ended up with these numbers. Let me explain.

The game I analyzed is 9/6 Jacks or Better, which has the return of 99.54%. The pay schedule is 800/50/25/9/6/4/3/2/1 from top payline to bottom. The analysis is based on the perfect strategy.

Enumerate Potential Holdings for Achieving a Royal Flush

The only way to achieve a royal flush in Jacks or Better, or other video poker games, is by holding suited royal cards. Alternatively, you could discard all hands and hope for a magical outcome, like what my friend experienced. The table above summarizes all scenarios where royal cards should be held according to the perfect strategy for all 2,598,960 possible dealt hands. For example, there are 2,652 total dealt hands that should be played by holding QKA. Another example is that there are 285,360 dealt hands that need to be discarded entirely.

Note: 

1. There is no scenario where you should hold a suited Ace and Ten. So, if you play with the perfect strategy, you should never hold suited Ace and Ten in 9/6 Jacks or Better.
2. According to my calculations, in 9/6 Jacks or Better game, there are 84,360 hands out of the total 2,598,960 where you should discard all 5 cards. This accounts for approximately 3.25% of the hands.

Summarize the numbers

Here we adopt the notation Bob Dancer used in his books/strategy cards, and sumamrize the table above into the following table by grouping the ways of holding in to 6 categories – 5RF, 4RF, etc.

Category	Count	Odds (1/p)
5RF 4RF 3RF 2RF 1RF Discard all 5 cards	4 936 27492 204660 390828 285360	649740 2776.6667 94.5351 12.6989 6.6499 9.1077

In video poker, a 5RF refers to a Royal Flush, while a 4RF indicates a hand such as AJKQ of hearts and another 5, commonly known as four-to-a-royal. Other categories, like 3RF, follow the same concept. A 1RF means holding a single high card, and ‘Nothing’ means discarding the entire hand because no card is worth keeping.

There are a total of 2,598,960 possible ways to deal a hand of 5 cards from a 52-card deck. Among these, there are exactly 4 ways to get a dealt Royal Flush, making the odds of getting such a hand 1 in 649,740.

There are 936 ways to get a dealt hand where you should hold four-to-royal. To complete the Royal Flush. The odds of getting such a dealt hand are 1 in about 2,777 hands.

Note: For the 3RF category, the number of dealt hands containing 3-to-a-royal is more than 27,492 hands. However, in hands like AJK of hearts and K2 of spades, the correct play is to hold the high pair of Kings. Therefore, hands with 3 royal cards are not necessarily in this category. This is also true for the 2RF category.

Category	Odds (1/p)	Odds to draw a royal flush (1/p)	Combined Odds (1/p)
5RF 4RF 3RF 2RF 1RF Discard all 5 cards	649740 2776.6667 94.5351 12.6989 6.6499 9.1077	1 47 1081 16215 178365 1533939	649740 130503.3333 102192.4836 205912.9112 1186106.165 13970584.89

To determine the number of ways a Royal Flush (RF) can be completed from different starting hands, we need to consider each category:

RF5 (Dealt Royal Flush) When you are dealt a Royal Flush, you simply hold all the cards. The machine will auto-lock them for you. There is only 1 way to draw from the remaining deck, which is essentially not drawing at all.

RF4 (Four Cards to a Royal Flush) If you have four cards to a Royal Flush, there is a 1 in 47 chance to draw the remaining card needed. This is calculated as (binom{47}{1} = 47).

RF3 (Three Cards to a Royal Flush) If you have three cards to a Royal Flush, there are (binom{47}{2} = 1081) ways to draw the remaining two cards needed.

Other Categories The same idea applies to other categories. You calculate the number of ways to draw the remaining cards needed to complete the Royal Flush based on the number of cards you already have.

Then we have combined odds, this is the final result we are looking for!

RF5 (Dealt Royal Flush) The odds of completing a royal flush by sheer luck are 1 in 649,740 hands.

RF4 (4-to-Royal Flush) The odds of completing a royal flush by holding 4-to-royal and successfully drawing the needed card are 1 in about 130,503 hands (calculated as 2776.6667 * 47).

RF3 (3-to-Royal Flush) The odds of completing a royal flush by holding 3-to-royal and successfully drawing the 2 needed cards are 1 in about 102193 hands. Interestingly, this is the “easiest” way to draw a royal flush in Jacks or Better! Many experienced players have likely completed more royal flushes by holding 3 cards. Surprisingly, completing a royal flush by holding 4-to-a-royal (RF4) is actually harder than doing so with 3-to-a-royal (RF3).

RF0 (Discarding All Cards) The odds of connecting a royal flush by discarding all the cards are 1 in about 13,970,585 hands. This is extremely rare! It happens 1 in about 21.5 dealt royal flush cycle.

Overall odds of getting a royal

Category	Combined Odds (1/p)	Combined Odds
5RF 4RF 3RF 2RF 1RF Discard all 5 cards	649740 130503.3333 102192.4836 205912.9112 1186106.165 13970584.89	1.53908E-06 7.66264E-06 9.78546E-06 4.85642E-06 8.43095E-07 7.1579E-08
TOTAL		2.47583E-05 (1 in 40390.54745)

The overall odds of getting a royal flush in Jacks or Better, when summing up all the odds in the table, is 1 in approximately 40,391. Video poker players generally refer to this as 1 in 40,000 hands.

Now you have it, the odds of achieving a royal flush in 9/6 Jacks or Better, assuming optimal strategy.

In our next post, we’ll be analyzing the NSU (Not So Ugly) Deuces game.

Thanks for reading! If you found this article useful, please share it and leave a comment. Stay tuned and have a great day!