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UNDERSTANDING ROYAL FLUSH CYCLES IN VIDEO POKER

by Henry Tamburin

Henry Tamburin is the author of five books on casino gambling, featured blackjack writer for dozens of magazines and web sites, editor of the Blackjack Insider Newsletter, and host of www.smartgaming.com.

One of the most misunderstood concepts in video poker is "the royal flush cycle." Itís important if you want to be successful at video poker that you understand what this is, and how it can affect your bankroll.

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 a Full Pay Deuces Wild game itís 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 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, youíll average one royal. In other words, in any given one set of 40,000 hands, you could wind up with more than royal or, heaven forbid, possibly no royals. (Would you care to guess what 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). Donít worry Öyou (or I) donít have to take out our calculators because my friend and fellow video poker author Dan Paymar (creator of Optimum Video Poker software trainer) 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% chance of getting one royal or no royals after playing one cycle of 40,000 hands. Since the royal flush contributes 1.98% toward the overall 99.5% ER for jacks-or-better, your return between royals is only 97.5% (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 play video poker Ö 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 bettcha and it was painful. I keep records of all my playing sessions and the worse streak I ever had was about 135,000 hands without a royal flush. Even though that was painful (for my bankroll and me), I have friends who play video poker that have gone way more than 135,000 hands between royals (ouch!).

However, letís be optimistic and look at the other side of the curve. I recently played roughly 40,000 hands of 9/6 Jacks or Better (one cycle) in Las Vegas 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%. 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% chance of hitting one or more royal flushes in one cycle and only a 36.8% chance of hitting no royals (does that make you feel any better?).

Do you think itís 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%) 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% chance of getting no royals after 120,000 hands (3 cycles), and 1.8% 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 donít 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).

Note: For more information on video poker bankroll and playing strategies, I recommend Dan Paymarís Optimum Video Poker software program.

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