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# Royal flush cycles

20 July 2012

A reader sent me this question:

Can you explain what is meant by a royal flush cycle. I thought video poker machines were random and you can't predict when you are going to be dealt a royal?

A royal flush cycle is the mathematically calculated average number of hands it takes to hit a royal flush using perfect strategy. The number of hands in a royal flush cycle varies slightly from game to game. For Jacks-or-Better, the royal flush cycle is 40,391 hands, whereas for a full pay Deuces Wild game it's 45,282 hands (See Table 1). Why is there a difference in the number of hands? Because, in some games, the playing strategy calls for holding more two- or three-card royal flushes than in other games, therefore, you will get more royal flushes.

Table 1
Royal Flush Cycles

 Game Cycle Jacks-or-Better 40,391 Double Bonus 48,048 Double Double Bonus 40,782 Deuces Wild-Full Pay 45,282 Deuces Wild-NSU 43,456 Joker Wild-Kings-or-Better 46,214

Most players expect to hit one royal flush after playing roughly 40,000 hands. That is not necessarily the case. The math says on average you will hit a royal flush once in every 40,000 hands, which means for a whole bunch of sets of 40,000 hands, you'll average one royal. In other words, in any given one set of 40,000 hands, you could wind up with more than one royal or, heaven forbid, possibly no royals. (Would you care to guess the chances of the latter catastrophe occurring? Keep reading for the answer.)

There is a mathematical formula that you can use to calculate the probability of hitting any number of royal flushes in any number of cycles (the formula is called the Poisson Distribution after the French mathematician Simèon Poisson, who developed the formula in the 19th century to calculate the probability of rare events). Don't worry...you (or I) don't have to take out our calculators because my friend and fellow video poker author Dan Paymar (author of Video Poker Optimum Play) has done the work for us. The calculations yield the following results for one cycle of 40,000 hands (Jacks-or-Better):

Table 2
Probability of Hitting a
Royal in One Cycle

 # of Royals Probability None 36.8% 1 36.8% 2 18.4% 3 6.1% 4 1.5% 5 0.3% 6 or more 0.1%

Wow! If you look at the data in Table 2, it says that you have the same 36.8 percent chance of getting one royal or no royals after playing one cycle of 40,000 hands. Since the royal flush contributes 1.98 percent toward the overall 99.5 percent ER for jacks-or-better, your return between royals is only 97.5 percent (meaning that your bankroll will more than likely head south from one royal flush to the next one). This, dear readers, is why you must have enough bankroll to cover those times when you play many hands without hitting a royal.

Have I ever played one cycle and not hit a royal? You betcha, and it was painful. I keep records of all my playing sessions and the worst streak I ever had was about 85,000 hands without a royal flush. Even though that was painful (for my bankroll and me), I have friends that have gone way more than 85,000 hands between royals (ouch!).

However, let's be optimistic and look at the other side of the curve. I just came back from a trip to Las Vegas where I played roughly 40,000 hands of Jacks-or-Better (one cycle) and I hit three royal flushes. Was I lucky? According to Table 2, the chance of hitting three royals in one cycle of Jacks-or-Better is a paltry 6.1 percent. So yes, I would consider myself very lucky to have hit three royal flushes during this trip.

The data in the above table also leads to this conclusion: You have about a 63 percent chance of hitting one or more royal flushes in one cycle and only a 36.8 percent chance of hitting no royals. Does that make you feel any better?

Do you think it's impossible to play 200,000 hands of Jacks-or-better (five cycles), which many video poker aficionados consider to be the "long term," without hitting one measly royal flush? I hate to be the bearer of bad news, but according to the data in Table 3, you have a 1 chance in 140 (0.7 percent) of not hitting a royal flush even after playing 200,000 hands (or roughly 400 hours of play).

Table 3
Probability of No Royals after X Cycles

 # Cycles Probability of No Royals 1 36.8% 2 13.5% 3 5.0% 4 1.8% 5 0.7% 6 0.3% 7 0.09% 8 0.03%

The percentages in Table 3 are scary. You have a 5 percent chance of getting no royals after 120,000 hands (3 cycles), and 1.8 percent chance after 160,000 hands (5 cycles). Even though the chance of winding up without a single royal flush are slim, if it were to occur, it could be financially catastrophic (especially if you don't have enough bankroll to weather this remote, but still possible, outcome).

The reality for video poker players is this: In any one royal flush cycle of roughly 40,000 hands, there are no guarantees that you will hit exactly one royal flush. If you are lucky, you could get more than one royal flush, and if unlucky, you could wind up without any royals (this is why having enough bankroll to play video poker is very important).

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Henry Tamburin

Henry Tamburin is the author of the best-selling book, Blackjack: Take The Money and Run, editor of the Blackjack Insider e-Newsletter, and Lead Instructor for the Golden Touch Blackjack course. For a free 3-month subscription to his blackjack newsletter with full membership privileges, visit www.bjinsider.com/free. For details on the Golden Touch Blackjack course visit www.goldentouchblackjack.com or call 866/WIN-BJ21. For a free copy of his casino gambling catalog featuring over 50 products call 888/353-3234 or visit the Internet store at www.smartgaming.com.

#### Henry Tamburin Websites:

www.smartgaming.com

#### Books by Henry Tamburin:

Henry Tamburin
Henry Tamburin is the author of the best-selling book, Blackjack: Take The Money and Run, editor of the Blackjack Insider e-Newsletter, and Lead Instructor for the Golden Touch Blackjack course. For a free 3-month subscription to his blackjack newsletter with full membership privileges, visit www.bjinsider.com/free. For details on the Golden Touch Blackjack course visit www.goldentouchblackjack.com or call 866/WIN-BJ21. For a free copy of his casino gambling catalog featuring over 50 products call 888/353-3234 or visit the Internet store at www.smartgaming.com.

#### Henry Tamburin Websites:

www.smartgaming.com